Method and constructions for space-time codes for PSK constellations for spatial diversity in multiple-element antenna systems

ABSTRACT

General binary design criteria for PSK-modulated space-time codes are provided. For linear binary PSK (BPSK) codes and quadrature PSK (QPSK) codes, the rank (i.e., binary projections) of the unmodulated code words, as binary matrices over the binary field, is used as a design criterion. Fundamental code constructions for both quasi-static and time-varying channels are provided.

This application claims priority to U.S. Provisional patent applicationSerial No. 60/101,029, filed Sep. 18, 1998 for “Method and Constructionsfor Space-Time Codes for PSK Constellations”, and U.S. Provisionalpatent application Serial No. 60/144,559, filed Jul. 16, 1999 for“Method and Constructions for Space-Time Codes for PSK ConstellationsII”, both of which were filed by A. Roger Hammons, Jr. and Hesham ElGamal.

FIELD OF THE INVENTION

The invention relates generally to PSK-modulated space-time codes andmore specifically to using fundamental code constructions forquasi-static and time-varying channels to provide full spatial diversityfor an arbitrary number of transmit antennas.

BACKGROUND OF THE INVENTION

Recent advances in coding theory include space-time codes which providediversity in multi-antenna systems over fading channels with channelcoding across a small number of transmit antennas. For wirelesscommunication systems, a number of challenges arise from the harsh RFpropagation environment characterized by channel fading and co-channelinterference (CCI). Channel fading can be attributed to diffuse andspecular multipath, while CCI arises from reuse of radio resources.Interleaved coded modulation on the transmit side of the system andmultiple antennas on the receive side are standard methods used inwireless communication systems to combat time-varying fading and tomitigate interference. Both are examples of diversity techniques.

Simple transmit diversity schemes (in which, for example, a delayedreplica of the transmitted signal is retransmitted through a second,spatially-independent antenna and the two signals are coherentlycombined at the receiver by a channel equalizer) have also beenconsidered within the wireless communications industry as a method tocombat multipath fading. From a coding perspective, such transmitdiversity schemes amount to repetition codes and encourage considerationof more sophisticated code designs. Information-theoretic studies havedemonstrated that the capacity of multi-antenna systems significantlyexceeds that of conventional single-antenna systems for fading channels.The challenge of designing channel codes for high capacity multi-antennasystems has led to the development of “space-time codes,” in whichcoding is performed across the spatial dimension (e.g, antenna channels)as well as time. The existing body of work on space-time codes relatesto trellis codes and a block coded modulation scheme based on orthogonaldesigns. Example code designs that achieve full diversity for systemswith only a small number of antennas (L=2 and 3) are known for bothstructures, with only a relatively small number of space-time codesbeing known. Thus, a need exists for a methodology of generating andusing code constructions which allow systematic development of powerfulspace-time codes such as general constructions that provide fulldiversity in wireless systems with a large number of antennas.

The main concepts of space-time coding for quasi-static, flat Rayleighfading channels and the prior knowledge as to how to design them willnow be discussed. For the purpose of discussion, a source generates kinformation symbols from the discrete alphabet X, which are encoded bythe error control code C to produce code words of length N=nL_(t) overthe symbol alphabet Y. The encoded symbols are parsed among L_(t)transmit antennas and then mapped by the modulator into constellationpoints from the discrete complex-valued signaling constellation Ω fortransmission across a channel. The modulated streams for all antennasare transmitted simultaneously. At the receiver, there are L_(r) receiveantennas to collect the incoming transmissions. The received basebandsignals are subsequently decoded by the space-time decoder. Each spatialchannel (the link between one transmit antenna and one receive antenna)is assumed to experience statistically independent flat Rayleigh fading.Receiver noise is assumed to be additive white Gaussian noise (AWGN). Aspace-time code consists as discussed herein preferably of an underlyingerror control code together with the spatial parsing format.

Definition 1 An L×n space-time code of size M consists of an (Ln,M)error control code C and a spatial parser a that maps each code wordvector {overscore (c)} εC to an L×n matrix c whose entries are arearrangement of those of {overscore (c)}. The space-time code is saidto be linear if both C and σ are linear.

Except as noted to the contrary, a standard parser is assumed which maps

{overscore (c)}=(c ₁ ¹ , c ₁ ² , . . . ,c ₁ ^(Lt) , c ₂ ¹ ,c ₂ ² , . . .,c ₂ ^(Lt) , . . . ,c _(n) ¹ ,c _(n) ² , . . . ,c _(n) ^(Lt))εC

to the matrix $c = {\begin{bmatrix}c_{1}^{1} & c_{2}^{1} & \ldots & c_{n}^{1} \\c_{1}^{2} & c_{2}^{2} & \ldots & c_{n}^{2} \\\vdots & \vdots & ⋰ & \vdots \\c_{1}^{L_{t}} & c_{2}^{L_{t}} & \ldots & c_{n}^{L_{t}}\end{bmatrix}.}$

In this notation, it is understood that c_(t) ^(i) is the code symbolassigned to transmit antenna i at time t.

Let f:y→Ω be the modulator mapping function. Then s=f(c) is the basebandversion of the code word as transmitted across the channel. For thissystem, the following baseband model of the received signal ispresented: $\begin{matrix}{{y_{t}^{j} = {{\sum\limits_{i = 1}^{L_{t}}\quad {\alpha_{ij}\quad s_{t}^{i}\quad \sqrt{E_{s}}}} + n_{t}^{j}}},} & (1)\end{matrix}$

where y_(t) ^(j) is the signal received at antenna j at time t; α_(ij)is the complex path gain from transmit antenna i to receive antenna j;s_(t) ^(i)=f(c_(t) ^(i)) is the transmitted constellation pointcorresponding to c_(t) ^(i); and n_(t) ^(j) is the AWGN noise sample forreceive antenna j at time t. The noise samples are independent samplesof a zero-mean complex Gaussian random variable with variance N₀/2 perdimension. The fading channel is quasi-static in the sense that, duringthe transmission of n code word symbols across any one of the links, thecomplex path gains do not change with time t, but are independent fromone code word transmission to the next. In matrix notation,

{overscore (Y)}={square root over (E)} _(s) {overscore (A)}D _(c)+{overscore (N)},  (2)

where ${\overset{\_}{Y} = \begin{bmatrix}y_{1}^{1} & y_{2}^{1} & \ldots & y_{n}^{1} & y_{1}^{2} & y_{2}^{2} & \ldots & y_{n}^{2} & \ldots & y_{1}^{L_{r}} & y_{2}^{L_{r}} & \ldots & y_{n}^{L_{r}}\end{bmatrix}},{\overset{\_}{N} = \begin{bmatrix}n_{1}^{1} & n_{2}^{1} & \ldots & n_{n}^{1} & n_{1}^{2} & n_{2}^{2} & \ldots & n_{n}^{2} & \ldots & n_{1}^{L_{r}} & n_{2}^{L_{r}} & \ldots & n_{n}^{L_{r}}\end{bmatrix}},{\overset{\_}{A} = \begin{bmatrix}\alpha_{11} & \alpha_{21} & \ldots & \alpha_{L_{t}\quad 1} & \alpha_{12} & \alpha_{22} & \ldots & \alpha_{L_{t}\quad 2} & \ldots & \alpha_{1L_{r}} & \alpha_{2L_{r}} & \ldots & \alpha_{L_{t}\quad L_{r}}\end{bmatrix}},{D_{c} = {\begin{bmatrix}{f\quad (c)} & 0 & \ldots & 0 \\0 & {f\quad (c)} & \ldots & 0 \\\vdots & ⋰ & ⋰ & 0 \\0 & 0 & \ldots & {f\quad (c)}\end{bmatrix}_{L_{r}\quad L_{t} \times L_{r}\quad n}.}}$

Let code word c be transmitted. Then the pairwise error probability thatthe decoder prefers the alternate code word e to c is given by

P(c→e|{α _(ij)})=P(V<0|{α_(ij)}),

where V=||{overscore (A)}(D_(c)−D_(c))+{overscore (N)}||²−||{overscore(N)}||² is a Gaussian random variable with mean E{V}=||{overscore (A)}(D_(c)−D_(e))||² and variance Var{V}=2N₀E{V}. Thus, $\begin{matrix}{{P\quad \left( {V < 0} \middle| \left\{ \alpha_{ij} \right\} \right)} = {Q\quad \left( \frac{{\overset{\_}{A}\quad \left( {D_{c} - D_{e}} \right)}}{\sqrt{2N_{0}}} \right)}} & (3) \\{\quad {\leq {\frac{1}{2}\quad \exp {\left\{ {{- \frac{1}{4N_{0}}}{{\overset{\_}{A}\quad \left( {D_{c} - D_{e}} \right)}}^{2}} \right\}.}}}} & (4)\end{matrix}$

For the quasi-static, flat Rayleigh fading channel, equation (4) can bemanipulated to yield the fundamental bound: $\begin{matrix}{{P\quad \left( \left. c\rightarrow e \right. \middle| \left\{ \alpha_{ij} \right\} \right)} \leq \left( \frac{1}{\prod\limits_{i = 1}^{r}\quad \left( {1 + {\lambda_{i}\quad {E_{s}/4}N_{0}}} \right)} \right)^{L_{r}}} & (5) \\{\quad {{\leq \left( \frac{\eta \quad E_{s}}{4N_{0}} \right)^{- {rL}_{r}}},}} & (6)\end{matrix}$

where r=rank(f(c)−f(e)) and η=(λ₁λ₂ . . . λ_(r))^(1/r) is the geometricmean of the nonzero eigenvalues of A=(f(c)−f(e))(f(c)−f(e))^(H).

This leads to the rank and equivalent product distance criteria forspace-time codes.

(1) Rank Criterion: Maximize the diversity advantage r=rank(f(c)−f(e))over all pairs of distinct code words c, e ε, and

(2) Product Distance Criterion: Maximize the coding advantage η=(λ₁λ₂ .. . λ_(r))^(1/r) over all pairs of distinct code words c, e ε.

The rank criterion is the more important of the two criteria as itdetermines the asymptotic slope of the performance curve as a functionof E_(s)/N₀. The product distance criterion is preferably of secondaryimportance and is ideally optimized after the diversity advantage ismaximized. For an L×n space-time code , the maximum possible rank is L.Consequently, full spatial diversity is achieved if all basebanddifference matrices corresponding to distinct code words in have fullrank L.

Simple design rules for space-time trellis codes have been proposed forL=2 spatial diversity as follows:

Rule 1. Transitions departing from the same state differ only in thesecond symbol.

Rule 2. Transitions merging at the same state differ only in the firstsymbol. When these rules are followed, the code word difference matricesare of the form${{f\quad (c)} - {f\quad (e)}} = \left\lbrack {\cdots \begin{matrix}x_{1} \\0\end{matrix}\cdots \begin{matrix}0 \\x_{2}\end{matrix}\cdots} \right\rbrack$

with x₁, x₂ nonzero complex numbers. Thus, every such difference matrixhas full rank, and the space-time code achieves 2-level spatialdiversity. Two good codes that satisfy these design rules, and a fewothers that do not, have been handcrafted using computer search methods.

The concept of “zeroes symmetry” has been introduced as a generalizationof the above-referenced design rules for higher levels of diversity L≧2.A space-time code has zeroes symmetry if every baseband code worddifference f(c)−f(e) is upper and lower triangular (and has appropriatenonzero entries to ensure full rank). The zeroes symmetry property issufficient for full rank but not necessary; nonetheless, it is useful inconstraining computer searches for good space-time codes.

Results of a computer search undertaken to identify full diversityspace-time codes with best possible coding advantage have beenpresented. A small table of short constraint length space-time trelliscodes that achieve full spatial diversity (L=2, 3, and 5 for BPSKmodulation; L=2 for QPSK modulation) is available. Difficulties,however, are encountered when evaluating diversity and coding advantagesfor general space-time trellis codes. As a general space-time codeconstruction, delay diversity schemes are known to achieve fulldiversity for all L≧2 with the fewest possible number of states.

A computer search similar to the above-referenced computer search hasidentified optimal L=2 QPSK space-time trellis codes of short constraintlength. The results agree with the previous results regarding theoptimal product distances but the given codes have different generators,indicating that, at least for L=2, there is a multiplicity of optimalcodes.

A simple transmitter diversity scheme for two antennas has beenintroduced which provides 2-level diversity gain with modest decodercomplexity. In this scheme, independent signalling constellation pointsx₁, x₂ are transmitted simultaneously by different transmit antennasduring a given symbol interval. On the next symbol interval, theconjugated signals −x₂* and x₁* are transmitted by the respectiveantennas. This scheme has the property that the two transmissions areorthogonal in both time and the spatial dimension.

The Hurwitz-Radon theory of real and complex orthogonal designs are aknown generalization this scheme to multiple transmit antennas.Orthogonal designs, however, are not space-time codes as defined hereinsince, depending on the constellation, the complex conjugate operationthat is essential to these designs may not have a discrete algebraicinterpretation. The complex generalized designs for L=3 and 4 antennasalso involve division by {square root over (2)}.

To summarize, studies on the problem of signal design for transmitdiversity systems have led to the development of the fundamentalperformance parameters for space-time codes over quasi-static fadingchannels such as: (1) diversity advantage, which describes theexponential decrease of decoded error rate versus signal-to-noise ratio(asymptotic slope of the performance curve on a log-log scale); and (2)coding advantage, which does not affect the asymptotic slope but resultsin a shift of the performance curve. These parameters are, respectively,the minimum rank and minimum geometric mean of the nonzero eigenvaluesamong a set of complex-valued matrices associated with the differencesbetween baseband modulated code words. A small number of interesting,handcrafted trellis codes for two antenna systems have been presentedwhich provide maximum 2-level diversity advantage and good codingadvantage.

One of the fundamental difficulties of space-time codes, which has sofar hindered the development of more general results, is the fact thatthe diversity and coding advantage design criteria apply to the complexdomain of baseband modulated signals, rather than to the binary ordiscrete domain in which the underlying codes are traditionallydesigned. Thus, a need also exists for binary rank criteria forgenerating BPSK and QPSK-modulated space-time codes.

SUMMARY OF THE INVENTION

The present invention overcomes the disadvantages of known trellis codesgenerated via design rules having very simple structure. In accordancewith the present invention, more sophisticated codes are provided usinga method involving design rules selected in accordance with thepreferred embodiment of the present invention. These codes arestraightforward to design and provide better performance than the knowncodes. The present invention also provides a significant advance in thetheory of space-time codes, as it provides a code design methodinvolving a powerful set of design rules in the binary domain. Currentdesign criteria are in the complex baseband domain, and the best codedesign rules to date are ad hoc with limited applicability.

The present invention further provides a systematic method, other thanthe simple delay diversity, of designing space-time codes to achievefull diversity for arbitrary numbers of antennas. The performance ofspace-time codes designed in accordance with the methodology andconstruction of the present invention exceed that of other knowndesigns.

Briefly summarized, the present invention relates to the design ofspace-time codes to achieve full spatial diversity over fading channels.A general binary design criteria for phase shift keying or PSK-modulatedspace-time codes is presented. For linear binary PSK (BPSK) codes andquadrature PSK (QPSK) codes, the rank (i.e., binary projections) of theunmodulated code words, as binary matrices over the binary field, is adesign criterion. Fundamental code constructions for both quasi-staticand time-varying channels are provided in accordance with the presentinvention.

A communication method in accordance with an embodiment of the presentinvention comprises the steps of generating information symbols for datablock frames of fixed length, encoding the generated information symbolswith an underlying error control code to produce the code word symbols,parsing the produced code word symbols to allocate the symbols in apresentation order to a plurality of antenna links, mapping the parsedcode word symbols onto constellation points from a discretecomplex-valued signaling constellation, transmitting the modulatedsymbols across a communication channel with the plurality of antennalinks, providing a plurality of receive antennas at a receiver tocollect incoming transmissions and decoding received baseband signalswith a space-time decoder.

BRIEF DESCRIPTION OF THE DRAWINGS

The various aspects, advantages and novel features of the presentinvention will be more readily comprehended from the following detaileddescription when read in conjunction with the appended drawings inwhich:

FIG. 1 is a block diagram of an exemplary digital cellular DirectSequence Code Division Multiple Access (DS-CDMA)base-station-to-mobile-station (or forward) link;

FIG. 2 is a block diagram of a system for a digital cellular systemwhich implements space-time encoding and decoding in accordance with anembodiment of the present invention;

FIG. 3 is a block diagram illustrating space-time encoding and decodingin accordance with an embodiment of the present invention;

FIG. 4 is a block diagram of a full-diversity space-time concatenatedencoder constructed in accordance with an embodiment of the presentinvention; and

FIGS. 5a, 5 b, 5 c and 5 d illustrate how known space-time codes forQPSK modulation over slow fading channels complies with general designrules selected in accordance with an embodiment of the presentinvention.

Throughout the drawing figures, like reference numerals will beunderstood to refer to like parts and components.

DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS

Referring to FIG. 1, by way of an example, a conventional digitalcellular Direct Sequence Code Division Multiple Access (DSCDMA)base-station-to-mobile-station (or forward) link 10 is shown using aconventional convolutional encoder and Viterbi decoder. FIG. 1 alsoillustrates the mobile-station-to-base-station (or reverse) link.

At the transmit end, the system 10 in FIG. 1 comprises a datasegmentation and framing module 16 where user information bits areassembled into fixed length frames from transmit data blocks 12. The Nbits per frame are input to the base station's convolutional encoder 18of rate r, which produces N/r code symbols at the input of the channelinterleaver 20. The channel interleaver 20 performs pseudo-randomshuffling of code symbols, and outputs the re-arranged symbols to thespread spectrum modulator 22. The spread spectrum modulator 22 uses auser-specific transmit PN-code generator 24 to produce a spread spectrumsignal which is carried on a RF carrier to the transmitter 26, where ahigh power amplifier coupled to the transmit antenna 28 radiates thesignal to the base station. The techniques of spread spectrum modulationand RF transmission are well known art to one familiar with spreadspectrum communications systems.

The signal received at the mobile station antenna 30 is amplified in theRF receiver 32 and demodulated by the spread spectrum demodulator 34,which uses the same PN-code generator 36 as used by the base stationtransmitter to de-spread the signal. The demodulated symbols arede-interleaved by the channel de-interleaver 38 and input to the Viterbidecoder 40. The decoded information bits are reconstructed using datablock reconstruction 42 into receive data blocks 14 and forwarded to thedata terminal equipment at the receive end.

With reference to FIG. 2, a digital cellularbase-station-to-mobile-station link is shown to illustrate theimplementation of space-time encoding and decoding in accordance with anembodiment of the present invention. While CDMA system is used as anexample, one familiar with the art would consider the present inventionapplicable to other types of wireless systems, which can employ othertypes of multiple access methods such as time division multiple access(TDMA).

Transmit data blocks 52 from the data terminal equipment are segmentedand framed 56 into fixed frame length and applied to the mobile'schannel space-time encoder 58. The output from a channel encoder 60 isfed to the space-time formatter 62 which determines the parsing(allocation and presentation order) of the coded symbols to the varioustransmit antennas 70 a, 70 b, 70 c. The spatial formatter output isapplied to the spread spectrum modulator 64 which uses a user specificPN-code generator 66 to create spread spectrum signals, carried on a RFcarrier via base RF transmitter 68, to the mobile station transmitter.The transmitter, with high power amplifier coupled to the Transmitantenna, radiates the signals via separate transmit antennas to themobile station.

The signal received at one or more mobile station antenna(s) 72 isamplified in the mobile RF receiver 74 and demodulated in a phase shiftkeying demodulator 76, which uses the same PN-code generator 78 as usedby the base station transmitter, to de-spread the signal. Thedemodulated symbols are processed at space-time decoder 80 by thespace-time de-formatter 82 and input to the channel decoder 84. Thedecoded information bits are reconstructed 86 into receive data blocks54 and forwarded to the data terminal equipment at the receive end.Based on the space-time code used, the de-formatter 82 and the decoder84 can be grouped in a single maximum likelihood receiver.

FIG. 3 illustrates an exemplary communication system 90 having a path 92from a source and a path 94 to a sink and which can be a system otherthan a cellular system. The system 90 has a space-time encoder 96 thatis similar to the encoder 58 depicted in FIG. 2 in that it comprises achannel encoder 98 and a spatial formatter 100. Plural modulators 102 a,102 b, 102 c, and so on, are also provided. At the receiver end, aspace-time demodulator 104 and a space-time decoder 106 are provided.

With continued reference to FIG. 3, the source generates k informationsymbols from a discrete alphabet X on the path 92 which are encoded byan error control code C by the space-time encoder 96. The space-timeencoder 96 produces code words of length N over the symbol alphabet Y.The encoded symbols are mapped by the modulators 102 a, 102 b, 102 c,and so on, onto constellation points from a discrete, complex-valuedsignaling constellation for transmission across the channel. Themodulated radio frequency signals for all of the L transmit antennas 102a, 102 b, 102 c, and so on, are transmitted at the same time to thereceiver space-time demodulator 104. The space-time channel decoder 106decodes the signals to the received data path 94. As shown, the receiverprovides M receive antennas to collect the incoming transmissions. Thereceived baseband signals are subsequently decoded by the space-timedecoder 106. The space-time code preferably includes an underlying errorcontrol code, together with the spatial parsing format as discussedbelow.

FIG. 4 depicts an exemplary concatenated space-time encoder 110 forimplementing a full- diversity space-time concatenated coding sequence.The coding sequence employs an outer code 112 which provides signals toa spatial formatter 114. Signals from the spatial formatter 114 areseparated for coding at inner code 116 a, 116 b, 116 c, and so on, whichprovide signals that are modulated, respectively, by modulators 118 a,118 b, 118 c, and so on, for transmission via antennas 120 a, 120 b, 120c, and so on. A convolutional encoder applying the binary rank criterionfor QPSK modulated space-time codes is shown in block diagram form inFIGS. 5a through 5 d in which known trellis space-time codes proposedfor QPSK modulation are shown to comply with the general design rules ofthe present invention. Space-time trellis codes are shown in FIGS. 5athrough 5 d, respectively, for 4, 8, 16, and 32 states which achievefull spatial diversity. As shown, the delay structures 122, 124, 126,and 128 provided for each respective code design are enough to ensurethat L=2 diversity is achieved. In the text below, a number of knowncodes are shown to be special cases of the general constructionspresented in accordance with the present invention. In addition, thepresent invention provides new delay diversity schemes and constructionssuch as examples of new BPSK space-time codes for L≧2 and new QPSKspace-time codes for L≧2.

The present invention is concerned primarily with the design ofspace-time codes rather than the signal processing required to decodethem. In most cases, the decoding employs known signal processing usedfor maximum likelihood reception.

The derivation of space-time codes from codes on graphs is a primaryfeature of the present invention, that is, to define constraints onmatrices for linear codes based on graphs to provide full spatialdiversity as space-time codes and therefore to design graphical codesfor space-time applications. The matrices can be obtained using thepresent invention. Graphical codes designed in this manner can bedecoded using soft-input, soft-output techniques. Thus, performance isclose to the Shannon limit. Accordingly, the code constructions ordesigns of the present invention define the state-of-the-art performanceof space-time codes. An improvement of iterative soft-input, soft-outputdecoding for a space-time channel is marginalization since the receiverneed only access the sum of the transmission from the L transmitantennas. This marginalization is improved via iteration.

A general stacking construction for BPSK and QPSK codes in quasi-staticfading channels is presented as another novel feature of the presentinvention. Examples of this construction are given by the rate 1/Lbinary convolutional codes for BPSK modulation. A preferred class ofQPSK modulated codes is the linear rate 1/L convolutional codes over theintegers modulo 4. Specific examples of selected block and concatenatedcoding schemes for L=2 and L=3 antennas with BPSK and QPSK modulationare provided below. In addition, a dyadic construction for QPSK signalsusing two binary full rank codes is also described below.

Another example is provided below of an expurgated, punctured version ofthe Golay code G₂₃ that can be formatted as a BPSK-modulated space-timeblock code achieving full L=2 spatial diversity and maximum bandwidthefficiency (rate 1 transmission). For L=3 diversity, an explicit rate 1space-time code is derived below which achieves full spatial diversityfor BPSK and QPSK modulation. By contrast, known space-time block codesderived from complex, generalized orthogonal designs provide no betterbandwidth efficiency than rate ¾.

The de-stacking construction is a method of obtaining good space-timeoverlays for existing systems for operation over time-varying fadingchannels. The key advantage of these systems is that of robustnessbecause they exploit time and space diversity. There is coding gain bothspatially (from the space-time “stacking”) and temporally (conventionalcoding gain achieved by “de-stacking”). The system is not dependententirely on the spatial diversity, which may not be available under alldeployment and channel circumstances. Examples of these are obtainedfrom de-stacking the rate 1/L convolution codes (BPSK) and (QPSK).

Multi-level code constructions with multi-stage decoding also follow thedesign criteria of the present invention. Since binary decisions aremade at each level, the BPSK design methodology of the present inventionapplies. For 8-PSK, the binary rank criteria developed for BPSK and QPSKcases also apply for the special case of L=2 antennas. This allows moresophisticated L=2 designs for PSK than is currently commerciallyavailable.

The design of space-time codes in accordance with the present inventionwill now be described. In Section 1, binary rank criteria for BPSK andQPSK-modulated space-time codes are discussed which are selected inaccordance with the present invention. Sections 2 and 3 expand on theuse of these criteria to develop comprehensive design criteria inaccordance with the present invention. In Section 2, new fundamentalconstructions for BPSK modulation are provided in accordance with thepresent invention that encompass such special cases as transmit delaydiversity schemes, rate 1/L convolutional codes, and certainconcatenated coding schemes. The general problem of formatting existingbinary codes into full-diversity space-time codes is also discussed.Specific space-time block codes of rate 1 for L=2 and L=3 antennas aregiven that provide coding gain, as well as achieve full spatialdiversity. In Section 3, ₄ analogs of the binary theory are provided inaccordance with the present invention. It is also shown that fulldiversity BPSK designs lift to full diversity QPSK designs. In Section4, the existing body of space-time trellis codes is shown to fit withinthe code design criteria of the present invention. Extension of thedesign criteria to time-varying channels is discussed in Section 5,which describes how multi-stacking constructions in accordance with thepresent invention provide a general class of “smart-greedy” space-timecodes for such channels. Finally, Section 6 discusses the applicabilityof the binary rank criteria to multi-level constructions forhigher-order constellations.

1 BINARY RANK CRITERIA FOR SPACE-TIME CODES

The design of space-time codes is hampered by the fact that the rankcriterion applies to the complex-valued differences between the basebandversions of the code words. It is not easy to transfer this designcriterion into the binary domain where the problem of code design isrelatively well understood. In section 1, general binary design criteriaare provided that are sufficient to guarantee that a space-time codeachieves full spatial diversity.

In the rank criterion for space-time codes, the sign of the differencesbetween modulated code word symbols is important. On the other hand, itis difficult to see how to preserve that information in the binarydomain. In accordance with the present invention, what can be said inthe absence of such specific structural knowledge is investigated byintroducing the following definition.

Definition 2 Two complex matrices r₁ and r₂ is said to be ω-equivalentif r₁ can be transformed into r₂ by multiplying any number of entries ofr₁ by powers of the complex number ω.

Interest primarily lies in the ω-equivalence of matrices when ω is agenerator for the signalling constellation Ω. Since BPSK and QPSK are ofparticular interest, the following special notation is introduced:${{{BPSK}\quad \left( {\omega = {- 1}} \right)\text{:}\quad r_{1}} \doteq {r_{2}\quad {denotes}\quad {that}\quad r_{1}\quad {and}\quad r_{2}\quad {are}\quad \left( {- 1} \right)\text{-}{{equivalent}.{QPSK}}\quad \left( {\omega = {i = \sqrt{- 1}}} \right)\text{:}\quad r_{1}}}\overset{..}{=}{r_{2}\quad {denotes}\quad {that}\quad r_{1}\quad {and}\quad r_{2}\quad {are}\quad i\text{-}{{equivalent}.}}$

Using this notion, binary rank criteria for space-time codes are derivedthat depend only on the unmodulated code words themselves. The binaryrank criterion provides a complete characterization for BPSK-modulatedcodes (under the assumption of lack of knowledge regarding signs in thebaseband differences). It provides a highly effective characterizationfor QPSK-modulated codes that, although not complete, provides a fertilenew framework for space-time code design.

The BPSK and QPSK binary rank criteria simplify the problem of codedesign and the verification that full spatial diversity is achieved.They apply to both trellis and block codes and for arbitrary numbers oftransmit antennas. In a sense, these results show that the problem ofachieving full spatial diversity is relatively easy. Within the largeclass of space-time codes satisfying the binary rank criteria, codedesign is reduced to the problem of product distance or coding advantageoptimization.

1.1 BPSK—Modulated Codes

For BPSK modulation, the natural discrete alphabet is the field ={0,1}of integers modulo 2. Modulation is performed by mapping the symbol x εto the constellation point s=f(x)ε{−1, 1} according to the rules=(−1)^(x). Note that it is possible for the modulation format toinclude an arbitrary phase offset e^(iφ), since a uniform rotation ofthe constellation will not affect the rank of the matrices f(c)−f(e) northe eigenvalues of the matrices A=(f(c)−f(e))(f(c)−f(e))^(H).Notationally, the circled operator ⊕ is used to distinguish modulo 2addition from real- or complex-valued (+, −) operations. It willsometimes be convenient to identify the binary digits 0, 1 ε with thecomplex numbers 0,1 ε. This is done herein without special comment ornotation.

Theorem 3 Let be a linear L×n space-time code with n≧L. Suppose thatevery non-zero binary code word c ε has the property that every realmatrix (−1)-equivalent to c is of full rank L. Then, for BPSKtransmission, satisfies the space-time rank criterion and achieves fullspatial diversity L.

Proof: It is enough to note that [(−1)^(c1)−(−1)^(c2)]/2{dot over(=)}c₁⊕c₂.

It turns out that (−1)-equivalence has a simple binary interpretation.The following lemma is used.

Lemma 4 Let M be a matrix of integers. Then the matrix equationM{overscore (x)}=0 has non-trivial real solutions if and only if it hasa non-trivial integral solution {overscore (x)}=[d₁,d₂, . . . , d_(L)]in which the integers d₁, d₂, . . . , d_(L) are jointly relativelyprime—that is, gcd(d₁, d₂, . . . , d_(L))=1.

Proof: Applying Gaussian elimination to the matrix M yields a canonicalform in which all entries are rational. Hence, the null space of M has abasis consisting of rational vectors. By multiplying and dividing byappropriate integer constants, any rational solution can be transformedinto an integral solution of the desired form.

Theorem 5 The L×n, (n≧L), binary matrix c=[ {overscore (c)}₁{overscore(c)}₂ . . . {overscore (c)}_(L)]^(T) has full rank L over the binaryfield if and only if every real matrix r=[{overscore (r)}₁ {overscore(r)}₂ . . . {overscore (r)}_(L)]^(T) that is (−1)-equivalent to c hasfull rank L over the real field .

Proof: (→) Suppose that r is not of full rank over . Then there existreal α₁, α₂, . . . , α_(L), not all zero, for which α₁{overscore(r)}₁+α₂{overscore (r)}₂+ . . . +α_(L){overscore (r)}_(L)=0. By thelemma, α_(i) are assumed to be integers and jointly relatively prime.Given the assumption on r and c, {overscore (r)}_(i)≡{overscore (c)}_(i)(mod 2). Therefore, reducing the integral equation modulo 2 produces abinary linear combination of the {overscore (c)}_(i) that sums to zero.Since the α_(i) are not all divisible by 2, the binary linearcombination is non-trivial. Hence, c is not of full rank over .

(→) Suppose that c is not of full rank over . Then there are rows{overscore (c)}_(i) ₁ , {overscore (c)}_(i) ₂ , . . . , {overscore(c)}_(i) _(ν) such that {overscore (c)}_(i) ₁ ⊕{overscore (c)}_(i) ₂ ⊕ .. . ⊕{overscore (c)}_(i) _(ν) ={overscore (0)}. Each column of ctherefore contains an even number of ones among these ν rows. Hence,the+and−signs in each column can be modified to produce a real-valuedsummation of these ν rows that is equal to zero. This modificationproduces a real-valued matrix that is (−1)-equivalent to c but is not offull rank.

The binary criterion for the design and selection of linear space-timecodes in accordance with the present invention now follows.

Theorem 6 (Binary Rank Criterion) Let be a linear L×n space-time codewith n≧L. Suppose that every non-zero binary code word c ε is a matrixof full rank over the binary field . Then, for BPSK transmission, thespace-time code achieves full spatial diversity L.

The binary rank criterion makes it possible to develop algebraic codedesigns for which full spatial diversity can be achieved withoutresorting to time consuming and detailed verification. Although thebinary rank criterion and of the present invention associated theoremsare stated for linear codes, it is clear from the proofs that they workin general, even if the code is nonlinear, when the results are appliedto the modulo 2 differences between code words instead of the code wordsthemselves.

1.2 QPSK—Modulated Codes

For QPSK modulation, the natural discrete alphabet is the ring ₄={0,±1,2} of integers modulo 4. Modulation is performed by mapping the symbol xε₄ to the constellation point s ε{±1,±i} according to the rule s=i^(x),where i={square root over (−1)}. Again, the absolute phase reference ofthe QPSK constellation can be chosen arbitrarily without affecting thediversity advantage or coding advantage of a ₄-valued space-time code.Notationally, subscripts are used to distinguish modulo 4 operations(⊕₄,⊖₄) from binary (⊕) and real- or complex-valued (+,−) operations.

For the ₄ valued matrix c, the binary component matrices α(c) and β(c)are defined to satisfy the expansion

c=β(c)+2α(c).

Thus, β(c) is the modulo 2 projection of c and α(c)=[c⊖₄β(c)]/2.

The following special matrices are now introduced which are useful inthe analysis of QPSK-modulated space-time codes:

(1) Complex-valued ζ(c)=c+iβ(c); and

(2) Binary-valued indicant projections: Ξ(c) and Ψ(c).

The indicant projections are defined based on a partitioning of c intotwo parts, according to whether the rows (or columns) are or are notmultiples of two, and serve to indicate certain aspects of the binarystructure of the ₄ matrix in which multiples of two are ignored.

A ₄-valued matrix c of dimension L×n is of type1^(l)2^(L−l)×1^(m)2^(n−m) if it consists of exactly l rows and m columnsthat are not multiples of two. It is of standard type1^(l)2^(L−l)×1^(m)2^(n−m) if it is of type 1^(l)2^(L−l)×1^(m)2^(n−m) andthe first l rows and first m columns in particular are not multiples oftwo. When the column (row) structure of a matrix is not of particularinterest, the matrix is of row type 1^(l)×2^(L−l) (column type1^(m)×2^(n−m)) or, more specifically, standard row (column) type.

Let c be a ₄-valued matrix of type 1^(l)2^(L−l)×1^(m)2^(n−m). Then,after suitable row and column permutations if necessary, it has thefollowing row and column structure: $c = {\begin{bmatrix}{\overset{\_}{c}}_{1} \\{\overset{\_}{c}}_{2} \\\vdots \\{\overset{\_}{c}}_{l} \\{2{\overset{\_}{c}}_{l + 1}^{\prime}} \\{2{\overset{\_}{c}}_{l + 2}^{\prime}} \\\vdots \\{2{\overset{\_}{c}}_{L}^{\prime}}\end{bmatrix} = {\begin{bmatrix}{\overset{\_}{h}}_{1}^{T} & {\overset{\_}{h}}_{2}^{T} & \ldots & {\overset{\_}{h}}_{m}^{T} & {2{\overset{\_}{h}}_{m + 1}^{\prime T}} & {2{\overset{\_}{h}}_{m + 2}^{\prime T}} & \ldots & {2{\overset{\_}{h}}_{n}^{\prime T}}\end{bmatrix}.}}$

Then the row-based indicant projection (Ξ-projection) is defined as${\Xi \quad (c)} = {\begin{bmatrix}{\beta \quad \left( {\overset{\_}{c}}_{1} \right)} \\{\beta \quad \left( {\overset{\_}{c}}_{2} \right)} \\\vdots \\{\beta \quad \left( {\overset{\_}{c}}_{l} \right)} \\{\beta \quad \left( {\overset{\_}{c}}_{l + 1}^{\prime} \right)} \\{\beta \quad \left( {\overset{\_}{c}}_{l + 2}^{\prime} \right)} \\\vdots \\{\beta \quad \left( {\overset{\_}{c}}_{L}^{\prime} \right)}\end{bmatrix}.}$

and the column-based indicant projection (Ψ-projection) is defined as

Ψ(c)=[β({overscore (h)} ₁ ^(T))β({overscore (h)} ₂ ^(T)) . . .β({overscore (h)} _(m) ^(T))β({overscore (h)}′ ₊₁ ^(T))β({overscore(h)}′ _(m+2) ^(T)) . . . β({overscore (h)}′ _(n) ^(T))].

Note that

[Ψ(c)]^(T)=Ξ(c ^(T)).  (7)

The first result shows that the baseband difference of twoQPSK-modulated code words is directly related to the ₄-difference of theunmodulated code words.

Proposition 7 Let be a ₄ space-time code. For x, y ε_(l), leti^(x)−i^(y) denote the baseband difference of the correspondingQPSK-modulated signals. Then,

i ^(x) −i ^(y){umlaut over (=)}ζ(x⊖ ₄ y).

Furthermore, any complex matrix z=r+is that is (−1)-equivalent toi^(x)−i^(y) has the property that

r≡s≡β(x⊖ ₄ y)≡x⊕y(mod 2).

Proof: Any component of i^(x)−i^(y) can be written as

i ^(x) −i ^(y) =−i ^(y)·(1−i ^(δ)),

where δ=x⊖₄ y. Since ${1 - i^{\delta}} = \left\{ \begin{matrix}{0,} & {\delta = 0} \\{{{1 - i} = {{- i} \cdot \left( {1 + i} \right)}},} & {\delta = 1} \\{2,} & {\delta = 2} \\{{{1 + i} = {{- i} \cdot \left( {{- 1} + i} \right)}},} & {{\delta = {- 1}},}\end{matrix} \right.$

the entry i^(x)−i^(y) can be turned into the complex number (x ⊖₄y)+i(x⊕y) by multiplying by ±1 or ±i as necessary. Thus,

i ^(x) −i ^(y)=(x ⊖ ₄ y)+i(x⊕y)=ζ(x ⊖ ₄ y),

as claimed.

For (−1)-equivalence, multiplication by ±i is not allowed. Under thisrestriction, it is no longer possible to separate z into the terms x⊖₄ yand x⊕y so cleanly; the discrepancies, however, amount to additions ofmultiples of 2. Hence, if z=r+is {dot over (=)}i−i^(y), then r≡x⊖₄y (mod2) and s≡x⊕y (mod 2).

Theorem 8 Let be a linear, L×n (n≧L) space-time code over ₄. Supposethat every non-zero code word c ε has the property that every complexmatrix i-equivalent to ζ(c) is of full rank L. Then, for QPSKtransmission, satisfies the space-time rank criterion and achieves fullspatial diversity L.

Proof: Since is linear, the ₄-difference between any two code words isalso a code word. The result then follows immediately from the previousproposition.

The indicant projections of the ₄-valued matrix c provide a significantamount of information regarding the singularity of ζ(c) and any of itsi-equivalents. Thus, the indicants provide the basis for our binary rankcriterion for QPSK-modulated space-time codes.

Theorem 9 Let c=[{overscore (c)}₁ {overscore (c)}₂ . . . {overscore(c)}_(L)]^(T) be an ₄-valued matrix of dimension L×n, (n≧L). If therow-based indicant Ξ(c) or the column-based indicant Ψ(c) has full rankL over , then every complex matrix z that is i-equivalent to ζ(c) hasfull rank L over the complex field .

Proof: Proof for the row-based indicant will now be provided. The prooffor the column-based indicant is similar.

By rearranging the rows of c if necessary, any row that is a multiple of2 can be assumed to appear as one of the last rows of the matrix. Thus,there is an l for which β({overscore (c)}_(i))≠0 whenever 1≦i≦l andβ({overscore (c)}_(i))=0 for l<i<L. The first l rows is called the1-part of c; the last L−rows is called the 2-part.

Suppose that $z = {\begin{bmatrix}{\overset{\_}{z}}_{1} \\{\overset{\_}{z}}_{2} \\\vdots \\{\overset{\_}{z}}_{L}\end{bmatrix} = \begin{bmatrix}{{\overset{\_}{r}}_{1} + {i{\overset{\_}{s}}_{1}}} \\{{\overset{\_}{r}}_{1} + {i{\overset{\_}{s}}_{2}}} \\\vdots \\{{\overset{\_}{r}}_{L} + {i{\overset{\_}{s}}_{L}}}\end{bmatrix}}$

is singular and is i-equivalent to ζ(c). Then there exist complexnumbers α₁ =a ₁ +ib ₁,α₂ =a ₂ +ib ₂ , . . . ,α _(L) =a _(L) +ib _(L),not all zero, for which $\begin{matrix}{{{\alpha_{1}\quad {\overset{\_}{z}}_{1}} + {\alpha_{2}\quad {\overset{\_}{z}}_{2}} + \ldots + {\alpha_{L}\quad {\overset{\_}{z}}_{L}}} = {{{\sum\limits_{i = 1}^{L}\quad \left( {{a_{i}\quad {\overset{\_}{r}}_{i}} - {b_{i}\quad {\overset{\_}{s}}_{i}}} \right)} + {i\quad {\sum\limits_{i = 1}^{L}\quad \left( {{b_{i}\quad {\overset{\_}{r}}_{i}} + {a_{i}\quad {\overset{\_}{s}}_{i}}} \right)}}} = 0.}} & (8)\end{matrix}$

Without loss of generality, the a _(i) , b _(i) are assumed to beintegers having greatest common divisor equal to 1. Hence, there is anonempty set of coefficients having real or imaginary part an oddinteger. The coefficient α_(i) is said to be even or odd depending onwhether two is or is not a common factor of a_(i) and b_(i). It is saidto be of homogeneous parity if a_(i) and b_(i) are of the same parity;otherwise, it is said to be of heterogeneous parity.

There are now several cases to consider based on the nature of thecoefficients applied to the 1-part and 2-part of z.

Case (i): There is an odd coefficient of heterogeneous parity applied tothe 1-part of z.

In this case, taking the projection of (8) modulo 2,${{\sum\limits_{i = 1}^{l}\quad {\left( {{\beta \quad \left( a_{i} \right)} \oplus {\beta \quad \left( b_{i} \right)}} \right)\quad \beta \quad \left( {\overset{\_}{c}}_{i} \right)}} = 0},$

since β({overscore (r)}_(i))=β({overscore (s)}_(i))=β({overscore(c)}_(i)) by the proposition. By assumption, at least one of the binarycoefficients β(a _(i))⊕β(b _(i)) is nonzero. Hence, this is anon-trivial linear combination of the first l rows of Ξ(c), and so Ξ(c)is not of full rank over .

Case (ii): All of the nonzero coefficients applied to the 1-part of zare homogeneous and at least one is odd; all of the coefficients appliedto the 2-part of z are homogeneous (odd or even).

In this case, equation (8) is multiplied by α*/2=(a−ib)/2, where α=a+ibis one of the coefficients applied to the 1-part of z having a and bboth odd. Note that α*α_(i) is even if α_(i) is homogeneous (odd oreven) and is odd homogeneous if α_(i) is heterogeneous. Hence, thisproduces a new linear combination, all coefficients of which still haveintegral real and imaginary parts. In this linear combination, one ofthe new coefficients is |α|²/2=(a ² +b ²)/2, which is an odd integer.The argument of case (i) now applies.

Case (iii): All of the nonzero coefficients applied to the 1-part of zare homogeneous and at least one is odd; there is a heterogeneouscoefficient applied to the 2-part of z.

In this case, normalization occurs as in case (ii), using one of the oddhomogeneous coefficients from the 1-part of z, say α=a+ib. Thus,normalization produces the equation

{tilde over (α)}₁ {overscore (z)} ₁+ . . . +{tilde over (α)}_(l){overscore (z)} _(l)+{tilde over (α)}_(l+1) {overscore (z)}′ _(l+1)+ . .. +{tilde over (α)}_(L) {overscore (z)}′ _(L)=0,  (9)

where {tilde over (α)}_(i)=α*α_(i)/2 for i≦l and {tilde over(α)}_(i)=α*α_(i) for i>l.

Taking the projection modulo 2 of the real (or imaginary) part ofequation (9) yields $\begin{matrix}{{0 = \quad {\sum\limits_{i = 1}^{l}\quad {{\left\lbrack {{\beta \quad \left( {\overset{\sim}{a}}_{i} \right)} \oplus {\beta \quad \left( {\overset{\sim}{b}}_{i} \right)}} \right\rbrack \beta \quad \left( {\overset{\sim}{c}}_{i} \right)} \oplus {\sum\limits_{i = {l + 1}}^{L}\quad {{\beta \quad \left( {\overset{\sim}{a}}_{i} \right)\quad \beta \quad \left( {\overset{\_}{r}}_{i}^{\prime} \right)} \oplus {\sum\limits_{i = {l + 1}}^{L}\quad {\beta \quad \left( {\overset{\sim}{b}}_{i} \right)\quad \beta \quad \left( {\overset{\_}{s}}_{i}^{\prime} \right)}}}}}}}\quad} \\{= \quad {\sum\limits_{i = 1}^{l}\quad {{\left\lbrack {{\beta \quad \left( {\overset{\sim}{a}}_{i} \right)} \oplus {\beta \quad \left( {\overset{\sim}{b}}_{i} \right)}} \right\rbrack \beta \quad \left( {\overset{\sim}{c}}_{i} \right)} \oplus {\sum\limits_{i = {l + 1}}^{L}\quad {\beta \quad \left( {\overset{\sim}{a}}_{i} \right)\quad {\left( {{\beta \quad \left( {\overset{\_}{r}}_{i}^{\prime} \right)} \oplus {\beta \quad \left( {\overset{\_}{s}}_{i}^{\prime} \right)}} \right).}}}}}}\end{matrix}$

For i≧l+1, it is true that β({overscore (r)}′_(i))⊕β({overscore(s)}′_(i))=β({overscore (c)}′_(i)), where {overscore(c)}_(i)=2{overscore (c)}′_(i) is the i-th row of c. By assumption,there is a nonzero coefficient in each of the three component sums.Hence, equation (9) establishes a nontrivial linear combination of therows of Ξ(c).

Case (iv): All of the coefficients applied to the 1-part of z are even,and at least one of the coefficients applied to the 2-part of z isheterogeneous.

In this case, equation (8) is divided by two to get the modifieddependence relation

α′₁ {overscore (z)} ₁+ . . . +α′_(l){overscore(z)}_(l)+α_(l+1){overscore (z)}′_(l+1)+ . . . +α_(L) {overscore (z)}′_(L)=0,  (10)

where α′_(i)=α_(i)/2 and {overscore (z)}′_(i)={overscore (z)}_(i)/2.Projecting modulo 2 gives two independent binary equations correspondingto the real and imaginary parts of equation (10):${{\sum\limits_{i = 1}^{l}\quad {{\left\lbrack {{\beta \quad \left( a_{i}^{\prime} \right)} \oplus {\beta \quad \left( b_{i}^{\prime} \right)}} \right\rbrack \beta \quad \left( {\overset{\_}{c}}_{i} \right)} \oplus {\sum\limits_{i = {l + 1}}^{L}\quad {{\beta \quad \left( a_{i} \right)\quad \beta \quad \left( {\overset{\_}{r}}_{i}^{\prime} \right)} \oplus {\sum\limits_{i = {l + 1}}^{L}\quad {\beta \quad \left( b_{i} \right)\quad \beta \quad \left( {\overset{\_}{s}}_{i}^{\prime} \right)}}}}}} = 0}\quad$${\sum\limits_{i = 1}^{l}\quad {{\left\lbrack {{\beta \quad \left( a_{i}^{\prime} \right)} \oplus {\beta \quad \left( b_{i}^{\prime} \right)}} \right\rbrack \beta \quad \left( {\overset{\_}{c}}_{i} \right)} \oplus {\sum\limits_{i = {l + 1}}^{L}\quad {{\beta \quad \left( b_{i} \right)\quad \beta \quad \left( {\overset{\_}{r}}_{i}^{\prime} \right)} \oplus {\sum\limits_{i = {l + 1}}^{L}\quad {\beta \quad \left( a_{i} \right)\quad \beta \quad \left( {\overset{\_}{s}}_{i}^{\prime} \right)}}}}}} = 0.$

Setting these two equal gives${{\sum\limits_{i = {l + 1}}^{L}\quad {\left\lbrack {{\beta \quad \left( a_{i} \right)} \oplus {\beta \quad \left( b_{i} \right)}} \right\rbrack \left\lbrack {{\beta \quad \left( {\overset{\_}{r}}_{i}^{\prime} \right)} \oplus {\beta \quad \left( {\overset{\_}{s}}_{i}^{\prime} \right)}} \right\rbrack}} = 0},$

which is a nontrivial linear combination of the rows β({overscore(c)}′_(i))=β({overscore (r)}′_(i))⊕β({overscore (s)}′_(i)), for i≦l+1,of Ξ(c).

Case (υ): All of the coefficents applied to the 1-part of z are even,and all of the coefficients applied to the two part of z arehomogeneous.

In this case, equation (10) is first used after dividing by two.Recalling that at least one of the coefficients α_(l+1), . . . α_(L) isodd, the modulo 2 projection of equation (10) is taken to get (fromeither the real or imaginary parts) the equation${\sum\limits_{i = 1}^{l}\quad {{\left\lbrack {{\beta \quad \left( a_{i}^{\prime} \right)} \oplus {\beta \quad \left( b_{i}^{\prime} \right)}} \right\rbrack \beta \quad \left( {\overset{\_}{c}}_{i} \right)} \oplus {\sum\limits_{i = {l + 1}}^{L}\quad {\beta \quad {\left( a_{i} \right)\quad\left\lbrack {{\beta \quad \left( {\overset{\_}{r}}_{i}^{\prime} \right)} \oplus {\beta \quad \left( {\overset{\_}{s}}_{i}^{\prime} \right)}} \right\rbrack}}}}} = 0.$

This is once again a nontrivial linear combination of the rows of Ξ(c).

The binary rank criterion for QPSK space-time codes in accordance withthe present invention now follows as an immediate consequence of theprevious two theorems.

Theorem 10 (QPSK Binary Rank Criterion I) Let be a linear L×n space-timecode over ₄, with n≧L. Suppose that, for every non-zero c ε, therow-based indicant Ξ(c) or the column-based indicant Ψ(c) has full rankL over . Then, for QPSK transmission, the space-time code achieves fullspatial diversity L.

In certain ₄ space-time code constructions, there may be no code wordmatrices having isolated rows or columns that are multiples of two. Forexample, it is possible for the entire code word to be a multiple oftwo. In this case, the following binary rank criterion is simpler yetsufficient.

Theorem 11 (QPSK Binary Rank Criterion II) Let be a linear L×nspace-time code over ₄, with n≧L. Suppose that, for every non-zero c ε,the binary matrix β(c) is of full rank over whenever β(c)≠0, and β(c/2)is of full rank over otherwise. Then, for QPSK transmission, thespace-time code achieves full spatial diversity L.

Proof: Under the specified assumptions, either Ξ(c)=β(c) or Ξ(c)=β(c/2),depending on whether β(c)=0 or not.

The QPSK binary rank criterion is a powerful tool in the design andanalysis of QPSK-modulated space-time codes.

2 THEORY OF BPSK SPACE-TIME CODES 2.1 Stacking Construction

A general construction for L×n space-time codes that achieve fullspatial diversity is given by the following theorem.

Theorem 12 (Stacking Construction) Let T₁, T₂, . . . , T_(L) be linearvector-space transformations from ^(k) into ^(n), and let be the L×nspace-time code of dimension k consisting of the code word matrices${{c\quad \left( \overset{\_}{x} \right)} = \begin{bmatrix}{T_{1}\quad \left( \overset{\_}{x} \right)} \\{T_{2}\quad \left( \overset{\_}{x} \right)} \\\vdots \\{T_{L}\quad \left( \overset{\_}{x} \right)}\end{bmatrix}},$

where {overscore (x)} denotes an arbitrary k-tuple of information bitsand n≦L. Then satisfies the binary rank criterion, and thus achievesfull spatial diversity L, if and only if T₁, T₂, . . . , T_(L) have theproperty that

∀a ₁ ,a ₂ , . . . , a _(L) ε:

T=a ₁T₁ ⊕a ₂T₂ ⊕ . . . ⊕a _(L)T_(L) is nonsingular unless a ₁ =a ₂ = . .. =a _(L)=0.

Proof: (→) Suppose satisfies the binary rank criterion but that T=a ₁T₁⊕a ₂T₂ ⊕ . . . ⊕a _(L)T_(L) is singular for some a₁, a₂, . . . , a_(L)ε. Then there is a non-zero {overscore (x)}₀ ε^(k) such thatT({overscore (x)}₀)=0. In this case,

T({overscore (x)}₀)=a ₁ ·T ₁({overscore (x)}₀)⊕a ₂ ·T ₂({overscore(x)}₀)⊕ . . . ⊕a _(L) ·T _(L)({overscore (x)}₀)=0

is a dependent linear combination of the rows of c({overscore (x)}₀) ε.Since satisfies the binary rank criterion, a₁=a₂= . . . =a_(L)=0.

(←) Suppose T₁, T₂, . . . , T_(L) have the stated property but thatc({overscore (x)}₀) ε is not of full rank. Then there exist a₁, a₂, . .. , a_(L) ε, not all zero, for which

T({overscore (x)}₀)=a ₁ ·T ₁({overscore (x)}₀)⊕a ₂ ·T ₂({overscore(x)}₀)⊕ . . . ⊕a _(L) ·T _(L)({overscore (x)}₀)=0,

where T=a ₁T₁ ⊕a ₂T₂⊕ . . . ⊕a _(L)T_(L). By hypothesis, T isnonsingular; hence, {overscore (x)} ₀=0 and c=0.

The vector-space transformations of the general stacking constructioncan be implemented as binary k×n matrices. In this case, the spatialdiversity achieved by the space-time code does not depend on the choiceof basis used to derive the matrices.

A heuristic explanation of the constraints imposed on the stackingconstruction will now be provided. In order to achieve spatial diversityL on a flat Rayleigh fading channel, the receiver is expected to be ableto recover from the simultaneous fading of any L−1 spatial channels andtherefore be able to extract the information vector {overscore (x)} fromany single, unfaded spatial channel (at least at high enoughsignal-to-noise ratio). This requires that each matrix M_(i) beinvertible. That each linear combination of the M_(i) must also beinvertible follows from similar reasoning and the fact that thetransmitted symbols are effectively summed by the channel.

The use of transmit delay diversity provides an example of the stackingconstruction. In this scheme, the transmission from antenna i is aone-symbol-delayed replica of the transmission from antenna i−1. Let Cbe a linear [n, k] binary code with (nonsingular) generator matrix G,and consider the delay diversity scheme in which code word {overscore(c)}={overscore (x)}G is repeated on each transmit antenna with theprescribed delay. The result is a space-time code achieving full spatialdiversity.

Theorem 13 Let be the L×(n+L−1) space-time code produced by applying thestacking construction to the matrices

M ₁ [G0_(k×(L−1)) ],M ₂=[0_(k×1) G0_(k×(L−2)) ]. . . ,M_(L)=[0_(k×(L−1)) G],

where 0 _(i×j) denotes the all-zero matrix consisting of i rows and jcolumns and G is the generator matrix of a linear [n, k] binary code.Then achieves full spatial diversity L.

Proof: In this construction, any linear combination of the M_(i) has thesame column space as that of G and thus is of full rank k. Hence, thestacking construction constraints are satisfied, and the space-time codeachieves full spatial diversity L.

A more sophisticated example of the stacking construction is given bythe class of binary convolutional codes. Let C be the binary, rate 1/L,convolutional code having transfer function matrix

G(D)=[g ₁(D)g ₂(D) . . . g _(L)(D)].

The natural space-time code associated with C is defined to consist ofthe code word matrices c(D)=G^(T)(D)x(D), where the polynomial x(D)represents the input information bit stream. In other words, for thenatural space-time code, the natural transmission format is used inwhich the output coded bits corresponding to g_(i)(x) are transmittedvia antenna i. The trellis codes are assumed to be terminated by tailbits. Thus, if x(D) is restricted to a block of N information bits, thenis an L×(N+ν) space-time code, where ν=max_(1≦i≦L){deg g_(i)(x)} is themaximal memory order of the convolutional code C.

Theorem 14 The natural space-time code associated with the rate 1/Lconvolutional code C satisfies the binary rank criterion, and thusachieves full spatial diversity L for BPSK transmission, if and only ifthe transfer function matrix G(D) of C has full rank L as a matrix ofcoefficients over .

Proof: Let g_(i)(D)=g_(i0)+g_(i1)D+g_(i2)D²+ . . . +g_(iν)D^(ν), wherei=1, 2, . . . , L. Then, the result follows from the stackingconstruction applied to the generator matrices ${M_{i} = \begin{bmatrix}g_{i0} & g_{i1} & \ldots & g_{iv} & 0 & \ldots & 0 \\0 & g_{i0} & g_{i1} & \ldots & g_{iv} & ⋰ & \vdots \\\vdots & ⋰ & ⋰ & ⋰ & ⋰ & ⋰ & 0 \\0 & \ldots & 0 & g_{i0} & g_{i1} & \ldots & g_{iv}\end{bmatrix}},$

each of which is of dimension N×(N+ν).

Alternately, Theorem 14 is proven by observing that${\sum\limits_{1 \leq i \leq L}^{\quad}\quad {a_{i}\quad g_{i}\quad (D)\quad x\quad (D)}} = 0$

for some${{x\quad (D)} \neq {0\quad {iff}\quad {\sum\limits_{1 \leq i \leq L}^{\quad}\quad {a_{i}\quad g_{i}\quad (D)}}}} = 0.$

This proof readily generalizes to recursive convolutional codes.

Since the coefficients of G(D) form a binary matrix of dimension L×(ν+1)and the column rank must be equal to the row rank, the theorem providesa simple bound as to how complex the convolutional code must be in orderto satisfy the binary rank criterion. It has been showed that the boundis necessary for the trellis code to achieve full spatial diversity.

Corollary 15 In order for the corresponding natural space-time code tosatisfy the binary rank criterion for spatial diversity L, a rate 1/Lconvolutional code C must have maximal memory order ν≧L−1.

Standard coding theory provides extensive tables of binary convolutionalcodes that achieve optimal values of free distance d_(free). Althoughthese codes are widely used in conventional systems, the formatting ofthem for use as space-time codes has not been studied previously. Asignificant aspect of the current invention is the separation of thechannel code from the spatial formatting and modulation functions sothat conventional channel codes can be adapted through use of the binaryrank criteria of the present invention to space-time communicationsystems. In Table I, many optimal rate 1/L convolutional codes arelisted whose natural space-time formatting in accordance with thepresent invention achieves full spatial diversity L. The table coversthe range of constraint lengths ν=2 through 10 for L=2, 3, 4 andconstraint lengths ν=2 through 8 for L=5, 6, 7, 8. Thus, Table Iprovides a substantial set of exemplary space-time codes of practicalcomplexity and performance that are well-suited for wirelesscommunication applications.

There are some gaps in Table I where the convolutional code with optimald_(free) is not suitable as a space-time code. It is straightforward,however, to find many convolutional codes with near-optimal d_(free)that satisfy the stacking construction of the present invention.

In the table, the smallest code achieving full spatial diversity L hasν=L rather than ν=L−1. This is because every optimal convolutional codeunder consideration for the table has all of its connection polynomialsof the form g_(i)(D)=1+ . . . +D^(ν); hence, the first and last columnsof G(D) are identical (all ones), so an additional column is needed toachieve rank L. Other convolutional codes of more general structure withν=L−1 could also be found using the techniques of the present invention.

TABLE I Binary Rate 1/L Convolutional Codes with Optimal d_(free) whoseNatural Space-Time Codes Achieve Full Spatial Diversity L v ConnectionPolynomials d_(free) 2  2 5, 7  5  3 64, 74  6  4 46, 72  7  5 65, 57  8 6 554, 744 10  7 712, 476 10  8 561, 753 12  9 4734, 6624 12 10 4672,7542 14 3  3 54, 64, 74 10  4 52, 66, 76 12  5 47, 53, 75 13  6 554,624, 764 15  7 452, 662, 756 16  8 557, 663, 711 18  9 4474, 5724, 715420 10 4726, 5562, 6372 22 4  4 52, 56, 66, 76 16  5 53, 67, 71, 75 18  7472, 572, 626, 736 22  8 463, 535, 733, 745 24  9 4474, 5724, 7154, 725427 10 4656, 4726, 5562, 6372 29 5  5 75, 71, 73, 65, 57 22  7 536, 466,646, 562, 736 28

In the stacking construction, the information vector {overscore (x)} isthe same for all transmit antennas. This is necessary to ensure fullrank in general. For example, if T₁(^(k))∩T₂(^(k))≠{{overscore (0)}},then the space-time code consisting of the matrices${c\quad \left( {\overset{\_}{x},\overset{\_}{y}} \right)} = \begin{bmatrix}{T_{1}\quad \left( \overset{\_}{x} \right)} \\{T_{2}\quad \left( \overset{\_}{y} \right)}\end{bmatrix}$

cannot achieve full spatial diversity even if T₁ and T₂ satisfy thestacking construction constraints. In this case choosing {overscore(x)}, {overscore (y)} so that T₁({overscore (x)})=T₂({overscore(y)})≠{overscore (0)} produces a code word matrix having two identicalrows. One consequence of this fact is that the natural space-time codesassociated with non-catastrophic convolutional codes of rate k/L withk>1 do not achieve full spatial diversity.

The natural space-time codes associated with certain Turbo codesillustrate a similar failure mechanism. In the case of a systematic,rate ⅓ turbo code with two identical constituent encoders, the all-oneinput produces an output space-time code word having two identical rows.

2.2 New Space-time Codes From Old

Transformations of space-time codes will now be discussed.

Theorem 16 Let be an L×m space-time code satisfying the binary rankcriterion. Given the linear vector-space transformation T:^(m)→^(n), anew L×n space-time code T() is constructed consisting of all code wordmatrices ${{T\quad (c)} = \begin{bmatrix}{T\quad \left( {\overset{\_}{c}}_{1} \right)} \\{T\quad \left( {\overset{\_}{c}}_{2} \right)} \\\vdots \\{T\quad \left( {\overset{\_}{c}}_{L} \right)}\end{bmatrix}},$

where c=[{overscore (c)}₁ {overscore (c)}₂ . . . {overscore(c)}_(L)]^(T)ε. Then, if T is nonsingular, T() satisfies the binary rankcriterion and, for BPSK transmission, achieves full spatial diversity L.

Proof: Let c, c′ ε, and consider the difference T(c)⊕T(c′)=T(Δc), whereΔc=c⊕c′=[Δ{overscore (c)}₁ Δ{overscore (c)}₂ . . . Δ{overscore(c)}_(L)]^(T)≠{overscore (0)}. Suppose

a ₁ T(Δ{overscore (c)}₁)⊕a ₂ T(Δ{overscore (c)}₂)⊕ . . . ⊕a _(L)T(Δ{overscore (c)}_(L))=0.

Then T(Δ{overscore (c)})=0 where Δ{overscore (c)}=a ₁Δ{overscore (c)}₁⊕a ₂Δ{overscore (c)}₂⊕ . . . ⊕a _(L)Δ{overscore (c)}_(L). Since T isnonsingular, Δ{overscore (c)}=0. But since satisfies the binary rankcriterion, a ₁ =a ₂= . . . =a _(L)=0.

Column transpositions applied uniformly to all code words in , forexample, do not affect the spatial diversity of the code. A moreinteresting interpretation of the theorem is provided by theconcatenated coding scheme of FIG. 4 in which T is a simple differentialencoder or a traditional [n, m] error control code that serves as acommon inner code for each spatial transmission.

Given two full-diversity space-time codes that satisfy the binary rankcriterion, they are combined into larger space-time codes that alsoachieve full spatial diversity. Let be a linear L×n_(A) space-time code,and let be a linear L×n_(B) space-time code, where L≦min{n_(A), n_(B)}.Their concatenation is the L×(n_(A)+n_(B)) space-time code ₁=|||consisting of all code word matrices of the form c=|a|b|, where a ε, bε.

A better construction is the space-time code ₂=||⊕| consisting of thecode word matrices c=|a|a⊕b|, where a ε b ε. (Zero padding is used toperform the addition if n_(A)≠n_(B)) Thus ₂ is anL×(n_(A)+max{n_(A),n_(B)}) space-time code.

The following proposition illustrates the full spatial diversity ofthese codes.

Theorem 17 The space-time codes ₁=||| and ₂=||⊕| satisfy the binary rankcriterion if and only if the space-time codes and do. As an applicationof the theorem, codes built according to the stacking construction canalso be “de-stacked.”

Theorem 18 (De-stacking Construction) Let be the L×n space-time code ofdimension k consisting of the code word matrices ${c = \begin{bmatrix}{\overset{\_}{x}\quad M_{1}} \\{\overset{\_}{x}\quad M_{2}} \\\vdots \\{\overset{\_}{x}\quad M_{L}}\end{bmatrix}},$

where M₁,M₂, . . . , M_(L) satisfy the stacking construction. Let l=L/pbe an integer divisor of L. Then the code _(l) consisting of code wordmatrices $c = \begin{bmatrix}{{\overset{\_}{x}}_{1}\quad M_{1}} & {{\overset{\_}{x}}_{2}\quad M_{l + 1}} & \ldots & {{\overset{\_}{x}}_{p}\quad M_{{{({p - 1})}l} + 1}} \\{{\overset{\_}{x}}_{1}\quad M_{2}} & {{\overset{\_}{x}}_{2}\quad M_{l + 2}} & \ldots & {{\overset{\_}{x}}_{p}\quad M_{{{{({p - 1})}l} + 2}\quad}} \\\vdots & \vdots & ⋰ & \vdots \\{{\overset{\_}{x}}_{1}\quad M_{l}} & {{\overset{\_}{x}}_{2}\quad M_{2l}} & \ldots & {{\overset{\_}{x}}_{p}\quad M_{L\quad}}\end{bmatrix}$

is an L/p×pn space-time code of dimension pk that achieves fulldiversity L/p. Setting {overscore (x)}₁={overscore (x)}₂= . . .={overscore (x)}_(p)={overscore (x)} produces an L/p×pn space-time codeof dimension k that achieves full diversity.

More generally, the following construction is provided in accordancewith the present invention.

Theorem 19 (Multi-stacking Construction) Let M={M₁, M₂, . . . , M_(L)}be a set of binary matrices of dimension k×n, n≧k, that satisfy thestacking construction constraints. For i=1, 2, . . . , m, let (M_(1i),M_(2i), . . . ,M_(li)) be an l-tuple of distinct matrices from the setM. Then, the space-time code consisting of the code words${c = \begin{bmatrix}{{\overset{\_}{x}}_{1}\quad M_{11}} & {{\overset{\_}{x}}_{2}\quad M_{12}} & \ldots & {{\overset{\_}{x}}_{m}\quad M_{1m}} \\{{\overset{\_}{x}}_{1}\quad M_{21}} & {{\overset{\_}{x}}_{2}\quad M_{22}} & \ldots & {{\overset{\_}{x}}_{m}\quad M_{2m}} \\\vdots & \vdots & ⋰ & \vdots \\{{\overset{\_}{x}}_{1}\quad M_{l1}} & {{\overset{\_}{x}}_{2}\quad M_{l2}} & \ldots & {{\overset{\_}{x}}_{m}\quad M_{{l\quad m}\quad}}\end{bmatrix}},$

is an l×mn space-time code of dimension mk that achieves full spatialdiversity l. Setting {overscore (x)}₁={overscore (x)}₂= . . .={overscore (x)}_(m)={overscore (x)} produces an l×mn space-time code ofdimension k that achieves full spatial diversity.

These modifications of an existing space-time code implicitly assumethat the channel remains quasi-static over the potentially longerduration of the new, modified code words. Even when this implicitassumption is not true and the channel becomes more rapidlytime-varying, however, these constructions are still of interest. Inthis case, the additional coding structure is useful for exploiting thetemporal as well as spatial diversity available in the channel. Section5 discusses this aspect of the invention.

2.3 Space-time Formatting of Binary Codes

Whether existing “time-only” binary error-correcting codes C can beformatted in a manner so as to produce a full-diversity space-time codeis now discussed. It turns out that the maximum achievable spatialdiversity of a code is not only limited by the code's least weight codewords but also by its maximal weight code words.

Theorem 20 Let C be a linear binary code of length n whose Hammingweight spectrum has minimum nonzero value d_(min) and maximum valued_(max). Then, there is no BPSK transmission format for which thecorresponding space-time code achieves spatial diversity L>min{d_(min),n−d_(max)+1}.

Proof: Let c be a code word of Hamming weight d=wt c. Then, in thebaseband difference matrix (−1)^(c)−(−1)⁰, between c and the all-zerocode word 0, the value −2 appears d times and the value 0 appears n−dtimes. Thus, the rank can be no more than d, since each independent rowmust have a nonzero entry, and can be no more than n−d+1 since theremust not be two identical rows containing only −2 entries. Therefore,the space-time code achieves spatial diversity at most${L \leq {\min\limits_{c \in c}\left\{ {{{wt}\quad c},{n - {{wt}\quad c} + 1}} \right\}}} = {\min {\left\{ {d_{\min},{n - d_{\max} + 1}} \right\}.}}$

This provides a general negative result useful in ruling out manyclasses of binary codes from consideration as space-time codes.

Corollary 21 If C is a linear binary code containing the all-1 codeword, then there is no BPSK transmission format for which thecorresponding space-time code achieves spatial diversity L>1. Hence, thefollowing binary codes admit no BPSK transmission format in which thecorresponding space-time code achieves spatial diversity L>1:

Repetition codes

Reed-Muller codes

Cyclic codes.

As noted in the discussion of the stacking construction, it is possibleto achieve full spatial diversity using repetition codes in a delaydiversity transmission scheme. This does not contradict the corollary,however, since the underlying binary code in such a scheme is notstrictly speaking a repetition code but a repetition code extended withextra zeros.

2.4 Exemplary Special Cases

In this section, special cases of the general theory for two and threeantenna systems are considered exploring alternative space-timetransmission formats and their connections to different partitionings ofthe generator matrix of the underlying binary code.

L=2 Diversity.

Let G=[I P] be a left-systematic generator matrix for a [2k, k] binarycode C, where I is the k×k identity matrix. Each code word row vector{overscore (c)}=({overscore (a)}_(I) {overscore (a)}_(p)) has first half{overscore (a)}_(I) consisting of all the information bits and secondhalf {overscore (a)}_(p) consisting of all the parity bits, where

{overscore (a)} _(p) ={overscore (a)} _(I) P.

Let be the space-time code derived from C in which the information bitsare transmitted on the first antenna and the parity bits are transmittedsimultaneously on the second antenna. The space-time code word matrixcorresponding to {overscore (c)}=({overscore (a)}_(I) {overscore(a)}_(p)) is given by $c = {\begin{bmatrix}{\overset{\_}{a}}_{I} \\{\overset{\_}{a}}_{P}\end{bmatrix}.}$

The following proposition follows immediately from the stackingconstruction theorem.

Proposition 22 If the binary matrices P and I⊕P are of full rank over ,then the space-time code achieves full L=2 spatial diversity. As anontrivial example of a new space-time block code achieving L=2 spatialdiversity, it is noted that both $P = \begin{bmatrix}1 & 0 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 \\0 & 1 & 1 & 1 & 0 & 1 & 1 & 0 & 0 & 0 & 1 \\1 & 1 & 1 & 0 & 1 & 1 & 1 & 0 & 0 & 1 & 0 \\1 & 1 & 0 & 1 & 1 & 1 & 0 & 0 & 1 & 0 & 1 \\0 & 1 & 1 & 1 & 1 & 0 & 0 & 1 & 0 & 1 & 1 \\1 & 1 & 1 & 1 & 0 & 0 & 0 & 0 & 1 & 1 & 0 \\1 & 1 & 1 & 0 & 0 & 0 & 1 & 1 & 1 & 0 & 1 \\1 & 1 & 0 & 0 & 0 & 1 & 0 & 1 & 0 & 1 & 1 \\0 & 1 & 0 & 0 & 1 & 0 & 1 & 0 & 1 & 1 & 1 \\0 & 1 & 0 & 1 & 0 & 1 & 1 & 1 & 1 & 1 & 0 \\0 & 1 & 1 & 0 & 1 & 1 & 0 & 1 & 1 & 0 & 0\end{bmatrix}$

and I⊕P are nonsingular over . Hence, the stacking construction producesa space-time of code achieving full L=2 spatial diversity. Theunderlying binary code C, with generator matrix G=[I|P], is anexpurgated and punctured version of the Golay code G₂₃. This is thefirst example of a space-time block code that achieves the highestpossible bandwidth efficiency and provides coding gain as well as fullspatial diversity.

The following proposition shows how to derive other L=2 space-time codesfrom a given one.

Proposition 23 If the binary matrix P satisfies the conditions of theabove theorem, so do the binary matrices P², P^(T), and UPU⁻¹, where Uis any change of basis matrix.

The (a|a+b) constructions are now reconsidered for the special case L=2.Let A and B be systematic binary [2k, k] codes with minimum Hammingdistances d_(A) and d_(B) and generator matrices G_(A)=[I P_(A)] andG_(B)=[I P_(B)], respectively. From the stacking construction, thecorresponding space-time codes and have code word matrices${c_{A} = \begin{bmatrix}{\overset{\_}{a}}_{I} \\{{\overset{\_}{a}}_{I}\quad P_{A}}\end{bmatrix}},{c_{B} = {\begin{bmatrix}{\overset{\_}{b}}_{I} \\{{\overset{\_}{b}}_{I}\quad P_{B}}\end{bmatrix}.}}$

The |a|a⊕b| construction produces a binary [4k,2k] code C with minimumHamming distance d_(c)=min{2 d _(A),d_(B)}. A nonsystematic generatormatrix for C is given by $G_{C} = {\begin{bmatrix}G_{A} & G_{A} \\0 & G_{B}\end{bmatrix} = {\begin{bmatrix}I & P_{A} & I & P_{A} \\0 & 0 & I & P_{B}\end{bmatrix}.}}$

Applying the stacking construction using the left and right halves ofG_(c) gives the space-time code =||⊕| of Theorem 22, in which the codeword matrices are of non-systematic form: $c_{C} = {\begin{bmatrix}{\overset{\_}{a}}_{I} & {{\overset{\_}{a}}_{I} \oplus {\overset{\_}{b}}_{I}} \\{{\overset{\_}{a}}_{I}\quad P_{A}} & {{{\overset{\_}{a}}_{I}\quad P_{A}} \oplus {{\overset{\_}{b}}_{I}\quad P_{B}}}\end{bmatrix}.}$

A systematic version is now derived in accordance with the presentinvention.

Proposition 24 Let and be 2×k space-time codes satisfying the binaryrank criterion. Let _(s) be the 2×2k space-time code consisting of thecode word matrices $c = {\begin{bmatrix}{\overset{\_}{a}}_{I} & {\overset{\_}{b}}_{I} \\{{\overset{\_}{a}}_{I}\quad P_{A}} & {{{\overset{\_}{a}}_{I}\quad P_{A}} \oplus {\left( {{\overset{\_}{a}}_{I} \oplus {\overset{\_}{b}}_{I}} \right)\quad P_{B}}}\end{bmatrix}.}$

Then _(s) also satisfies the binary rank criterion and achieves full L=2spatial diversity.

Proof: Applying Gaussian elimination to G_(c) and reordering columnsproduces the systematic generator matrix

G _(C) =[I _(2k×2k) P _(C)],

where $P_{C} = {\begin{bmatrix}P_{A} & {P_{A} \oplus P_{B}} \\0 & P_{B}\end{bmatrix}.}$

Note that P_(C) is nonsingular since P_(A) and P_(B) are bothnonsingular. Likewise,${I_{2k \times 2k} \oplus P_{C}} = \begin{bmatrix}{I \oplus P_{A}} & {P_{A} \oplus P_{B}} \\0 & {I \oplus P_{B}}\end{bmatrix}$

is nonsingular since I⊕P_(A) and I⊕P_(B) are. The rest follows from thestacking construction.

An alternate transmission format for 2×k space-time codes is nowconsidered. Let C be a linear, left-systematic [2k, k] code withgenerator matrix $G = \begin{bmatrix}I & 0 & A_{11} & A_{12} \\0 & I & A_{21} & A_{22}\end{bmatrix}$

where the submatrices I, 0, and A_(ij) are of dimension k/2×k/2. In thenew transmission format, the information vector is divided into twoparts {overscore (x)}₁ and {overscore (x)}₂ which are transmitted acrossdifferent antennas. Thus, the corresponding space-time code consists ofcode word matrices of the form ${c = \begin{bmatrix}{\overset{\_}{x}}_{1} & {\overset{\_}{p}}_{1} \\{\overset{\_}{x}}_{2} & {\overset{\_}{p}}_{2}\end{bmatrix}},$

where {overscore (p)}₁={overscore (x)}x₁A₁₁⊕{overscore (x)}₂A₂₁ and{overscore (p)}₂={overscore (x)}₁A₁₂⊕{overscore (x)}₂A₂₂.

For such codes, the following theorem gives sufficient conditions on thebinary connection matrices to ensure full spatial diversity of thespace-time code.

Proposition 25 Let A₁₂, A₂₁, and A=Σ_(i=1) ²(A_(i1)⊕A_(i2)) benon-singular matrices over . Then the space-time code achieves full L=2spatial diversity.

Proof: The conditions follow immediately from the stacking constructiontheorem applied to the matrices ${M_{1} = \begin{bmatrix}I & A_{11} \\0 & A_{21}\end{bmatrix}},{M_{2} = \begin{bmatrix}0 & A_{12} \\I & A_{22}\end{bmatrix}},$

since the sum M=M₁⊕M₂ may be reduced to the form $M = \begin{bmatrix}I & {A_{11} \oplus A_{12}} \\0 & A\end{bmatrix}$

by Gaussian elimination.

The conditions of the proposition are not difficult to satisfy. Forexample, consider the linear 2×4 space-time code whose code words$c = \begin{bmatrix}x_{11} & x_{12} & p_{11} & p_{12} \\x_{21} & x_{22} & p_{21} & p_{22}\end{bmatrix}$

are governed by the parity check equations p₁₁ = x₁₂ ⊕ x₂₁ ⊕ x₂₂p₁₂ = x₁₂ ⊕ x₂₂ p₂₁ = x₁₁ ⊕ x₂₁ p₂₂ = x₁₁ ⊕ x₁₂ ⊕ x₂₁.

The underlying binary code C has a generator matrix with submatrices$A_{11} = \begin{bmatrix}0 & 0 \\1 & 1\end{bmatrix}$ $A_{12} = \begin{bmatrix}1 & 1 \\0 & 1\end{bmatrix}$ $A_{21} = \begin{bmatrix}1 & 0 \\1 & 1\end{bmatrix}$ ${A_{22} = \begin{bmatrix}1 & 1 \\0 & 0\end{bmatrix}},$

which meet the requirements of the proposition. Hence, achieves 2-levelspatial diversity. L=3 Diversity.

Similar derivations for L=3 antennas are straightforward. The followingexample is interesting in that it provides maximum possible bandwidthefficiency (rate 1 transmission) while attaining full spatial diversityfor BPSK or QPSK modulation. The space-time block codes derived fromcomplex generalized orthogonal designs for L>2, on the other hand,achieve full diversity only at a loss in bandwidth efficiency. Theproblem of finding generalized orthogonal designs of rates greater than¾ for L>2 is a difficult problem. Further, rate 1 space-time block codesof short length can not be designed by using the general method of delaydiversity. By contrast, the following rate 1 space-time block code forL=3 is derived by hand.

Let consist of the code word matrices ${c = \begin{bmatrix}{\overset{\_}{x}\quad M_{1}} \\{\overset{\_}{x}\quad M_{2}} \\{\overset{\_}{x}\quad M_{3}}\end{bmatrix}},$

where ${M_{1} = \begin{bmatrix}1 & 0 & 0 \\0 & 1 & 0 \\0 & 0 & 1\end{bmatrix}},{M_{2} = \begin{bmatrix}0 & 0 & 1 \\1 & 0 & 1 \\0 & 1 & 0\end{bmatrix}},{M_{3} = {\begin{bmatrix}0 & 1 & 0 \\0 & 1 & 1 \\1 & 0 & 1\end{bmatrix}.}}$

It is easily verified that M₁, M₂, M₃ satisfy the stacking constructionconstraints. Thus, is a 3×3 space-time code achieving full spatialdiversity (for QPSK as well as BPSK transmission). Since admits a simplemaximum likelihood decoder (code dimension is three), it can be used asa 3-diversity space-time applique for BPSK- or QPSK-modulated systemssimilar to the 2-diversity orthogonal design scheme.

Similar examples for arbitrary L>3 can also be easily derived. Forexample, matrix M₁ can be interpreted as the unit element in the Galoisfield GF(2 ³); M₂ as the primitive element in GF(2 ³) satisfying α³=1+α;and M₃ as its square α². Since 1, α, α² are linear independent over ,the BPSK stacking construction of the current invention is satisfied.Any set of linearly independent elements from GF(2 ³) can be similarlyexpressed as a set of 3×3 matrices satisfying the BPSK stackingconstruction and hence would provide other examples of L=3 full spatialdiversity space-time block codes in accordance with the teachings of thepresent invention. This construction method extends to an arbitrarynumber L of transmit antennas by selecting a set of L linearlyindependent elements in GF(2 ^(L)).

3 THEORY OF QPSK SPACE-TIME CODES

Due to the binary rank criterion developed for QPSK codes, the richtheory developed in section 2 for BPSK-modulated space-time codeslargely carries over to QPSK modulation. Space-time codes for BPSKmodulation are of fundamental importance in the theory of space-timecodes for QPSK modulation.

3.1 ₄ Stacking Constructions

The binary indicant projections allow the fundamental stackingconstruction for BPSK-modulated space-time codes to be “lifted” to thedomain of QPSK-modulated space-time codes.

Theorem 26 Let M₁, M₂, . . . , M_(L) be ₄-valued m×n matrices ofstandard row type 1 ^(l) 2 ^(m−l) having the property that

∀a ₁ ,a ₂ , . . . , a _(L)ε:

a ₁Ξ(M ₁)⊕a ₂Ξ(M ₂)⊕ . . . ⊕a _(LΞ(M) _(L)) is nonsingular

unless a ₁ =a ₂ = . . . a _(L)=0.

Let be the L×n space-time code of size M=2 ^(l+m) consisting of allmatrices${{c\quad \left( {\overset{\_}{x},\overset{\_}{y}} \right)} = \begin{bmatrix}{\left( {\overset{\_}{x}\quad \overset{\_}{y}} \right)\quad M_{1}} \\{\left( {\overset{\_}{x}\quad \overset{\_}{y}} \right)\quad M_{2}} \\\vdots \\{\left( {\overset{\_}{x}\quad \overset{\_}{y}} \right)\quad M_{L}}\end{bmatrix}},$

where ({overscore (x y)}) denotes an arbitrary indexing vector ofinformation symbols {overscore (x)}ε₄ ^(l) and {overscore (y)}ε ^(m−l).Then, for QPSK transmission, satisfies the QPSK binary rank criterionand achieves full spatial diversity L.

Proof: Suppose that for some {overscore (x)}₀,{overscore (y)}₀, not bothzero, the code word c({overscore (x)}₀,{overscore (y)}₀) hasΞ-projection not of full rank over . It must be shown that the matricesM_(i) do not have the stated nonsingularity property.

 Case(i):β({overscore (x)} ₀)≠0.

If there are rows of c that are multiples of two, the failure of theM_(i) to satisfy the nonsingularity property is easily seen. In thiscase, there is some row l of c for which

0=β(({overscore (x)} ₀ {overscore (y)} ₀)M _(l))=(β({overscore (x)}_(o)){overscore (0)})Ξ(M _(l)).

Hence, Ξ(M_(l)) is singular, establishing the desired result.

Therefore, c is assumed to have no rows that are multiples of two, sothat Ξ(c)=β(c). Then there exist a₁, a₂, . . . , a_(L) ε, not all zero,such that $\begin{matrix}{0 = \quad {{a_{1}\quad \beta \quad \left( {{\overset{\_}{x}}_{0}\quad M_{1}} \right)} \oplus {a_{2}\quad \beta \quad \left( {{\overset{\_}{x}}_{0}\quad M_{2}} \right)} \oplus \ldots \oplus {a_{L}\quad \beta \quad \left( {{\overset{\_}{x}}_{0}\quad M_{L}} \right)}}} \\{= \quad {\beta \quad \left( {\overset{\_}{x}}_{0} \right)\quad {\left( {{a_{1}\quad \Xi \quad \left( M_{1} \right)} \oplus {a_{2}\quad \Xi \quad \left( M_{2} \right)} \oplus \ldots \oplus {a_{L}\quad \Xi \quad \left( M_{L} \right)}} \right).}}}\end{matrix}$

Since β({overscore (x)}₀)≠0, a ₁Ξ(M₁)⊕a ₂Ξ(M₂)⊕ . . . ⊕a _(L)Ξ(M_(L)) issingular, as was to be shown.

Case (ii):β({overscore (x)}₀)=0.

In this case, all of the rows of c are multiples of two. Letting${c = \begin{bmatrix}{\left( {2{\overset{\_}{x}}_{0}^{\prime}\quad {\overset{\_}{y}}_{0}} \right)\quad M_{1}} \\{\left( {2{\overset{\_}{x}}_{0}^{\prime}\quad {\overset{\_}{y}}_{0}} \right)\quad M_{2}} \\\vdots \\{\left( {2{\overset{\_}{x}}_{0}^{\prime}\quad {\overset{\_}{y}}_{0}} \right)\quad M_{L}}\end{bmatrix}},$

where {overscore (x)}′₀ ε^(l), then${\Xi \quad (c)} = {\begin{bmatrix}{\left( {{\overset{\_}{x}}_{0}^{\prime}\quad {\overset{\_}{y}}_{0}} \right)\quad \Xi \quad \left( M_{1} \right)} \\{\left( {{\overset{\_}{x}}_{0}^{\prime}\quad {\overset{\_}{y}}_{0}} \right)\quad \Xi \quad \left( M_{2} \right)} \\\vdots \\{\left( {{\overset{\_}{x}}_{0}^{\prime}\quad {\overset{\_}{y}}_{0}} \right)\quad \Xi \quad \left( M_{L} \right)}\end{bmatrix}.}$

By hypothesis, there exist a ₁ , a ₂ , . . . , a _(L) ε, not all zero,such that

a ₁·({overscore (x)}′ ₀ {overscore (y)} ₀)Ξ(M ₁)⊕a ₂·({overscore (x)}′ ₀{overscore (y)} ₀)Ξ(M ₂)⊕ . . . ⊕a _(L)·({overscore (x)}′ ₀ {overscore(y)} ₀)Ξ(M _(L))=0.

Then a ₁Ξ(M₁)⊕a ₂Ξ(M₂)⊕ . . . ⊕a _(L)Ξ(M_(L)) is singular as was to beshown.

In summary, the stacking of ₄-valued matrices produces a QPSK-modulatedspace-time code achieving full spatial diversity if the stacking oftheir Ξ-projections produces a BPSK-modulated space-time code achievingfull diversity. Thus, the binary constructions lift in a natural way.Analogs of the transmit delay diversity construction, rate 1/Lconvolutional code construction, ||⊕| construction, and multi-stackingconstruction all follow as immediate consequences of the QPSK stackingconstruction and the corresponding results for BPSK-modulated space-timecodes.

Theorem 27 Let be the ₄-valued, L×(n+L−1) space-time code produced byapplying the stacking construction to the matrices

M ₁ =[G0_(k×(L−1)) ],M ₂=[0_(k×1) G0_(k×(L−2)) ], . . . , M_(L)=[0_(k×(L−1)) G],

where 0_(ixj) denotes the all-zero matrix consisting of i rows and jcolumns and G is the generator matrix of a linear ₄-valued code oflength n. If Ξ(G) is of full rank over , then the QPSK-modulated codeachieves full spatial diversity L.

Theorem 28 The natural space-time code associated with the rate 1/Lconvolutional code C over ₄ achieves full spatial diversity L for QPSKtransmission if the transfer function matrix G(D) of C has Ξ-projectionof full rank L as a matrix of coefficients over .

Theorem 29 The ₄-valued space-time codes ₁=||| and ₂=||⊕| satisfy theQPSK binary rank criterion if and only if the ₄-valued space-time codesand do.

Theorem 30 Let be the L×n space-time code of size M=2^(u+m) consistingof the code word matrices ${c = \begin{bmatrix}{\left( {\overset{\_}{x}\quad \overset{\_}{y}} \right)\quad M_{1}} \\{\left( {\overset{\_}{x}\quad \overset{\_}{y}} \right)\quad M_{2}} \\\vdots \\{\left( {\overset{\_}{x}\quad \overset{\_}{y}} \right)\quad M_{L}}\end{bmatrix}},$

where {overscore (x)}ε₄ ^(u), {overscore (y)}ε^(m−u), and the ₄-valuedM₁,M₂, . . . , M_(L) of standard row type 1 ^(u) 2 ^(m−u) satisfy thestacking construction constraints for QPSK-modulated codes. For i=1, 2,. . . , m, let (M_(1i), M_(2i), . . . , M_(li)) be an l-tuple ofdistinct matrices from the set {M₁, M₂, . . . , M_(L)}. Then, thespace-time code _(l,m) consisting of the code words $c = \begin{bmatrix}{\left( {{\overset{\_}{x}}_{1}\quad {\overset{\_}{y}}_{1}} \right)\quad M_{11}} & {\left( {{\overset{\_}{x}}_{2}\quad {\overset{\_}{y}}_{2}} \right)\quad M_{12}} & \cdots & {\left( {{\overset{\_}{x}}_{m}\quad {\overset{\_}{y}}_{m}} \right)\quad M_{1m}} \\{\left( {{\overset{\_}{x}}_{1}\quad {\overset{\_}{y}}_{1}} \right)\quad M_{21}} & {\left( {{\overset{\_}{x}}_{2}\quad {\overset{\_}{y}}_{2}} \right)\quad M_{22}} & \cdots & {\left( {{\overset{\_}{x}}_{m}\quad {\overset{\_}{y}}_{m}} \right)\quad M_{2m}} \\\vdots & \vdots & ⋰ & \vdots \\{\left( {{\overset{\_}{x}}_{1}\quad {\overset{\_}{y}}_{1}} \right)\quad M_{l1}} & {\left( {{\overset{\_}{x}}_{2}\quad {\overset{\_}{y}}_{2}} \right)\quad M_{l2}} & \ldots & {\left( {{\overset{\_}{x}}_{m}\quad {\overset{\_}{y}}_{m}} \right)\quad M_{l\quad m}}\end{bmatrix}$

is an l×mn space-time code of size M^(m) that achieves full diversity l.Setting ({overscore (x)}₁ {overscore (y)}₁)=({overscore (x)}₂ {overscore(y)}₂) = . . . =({overscore (x)}_(m) {overscore (y)}_(m))=({overscore (xy)}) produces an l×mn space-time code of size M that achieves fulldiversity.

As a consequence of these results, for example, the binary connectionpolynomials of Table I can be used as one aspect of the currentinvention to generate linear, ₄-valued, rate 1/L convolutional codeswhose natural space-time formatting achieves full spatial diversity L.More generally, any set of ₄-valued connection polynomials whose modulo2 projections appear in the table can be used.

The transformation theorem also extends to QPSK-modulated space-timecodes in a straightforward manner.

Theorem 31 Let be a ₄-valued, L×m space-time code satisfying the QPSKbinary rank criterion with respect to Ξ-indicants, and let M be an m×n₄-valued matrix whose binary projection β(M) is nonsingular over .Consider the L×n space-time code M() consisting of all code wordmatrices ${{M\quad (c)} = \begin{bmatrix}{{\overset{\_}{c}}_{1}\quad M} \\{{\overset{\_}{c}}_{2}\quad M} \\\vdots \\{{\overset{\_}{c}}_{L}\quad M}\end{bmatrix}},$

where c=[{overscore (c)}₁ {overscore (c)}₂ . . . {overscore(c)}_(L)]^(T) ε: Then, M() satisfies the QPSK binary rank criterion andthus, for QPSK transmission, achieves full spatial diversity L.

Proof: Let c, c′ be distinct code words in , and let Δc=c⊖₄c′. Withoutloss of generality, Δc is assumed to be of standard row type 1 ^(l) 2^(L−l). Since β(M) is nonsingular, we have β({overscore (x)})β(M)=0 ifand only if β({overscore (x)})=0. Hence, M(Δc) is also of standard rowtype 1 ^(l) 2 ^(L−l). It is to be shown that Ξ(M(Δc)) is of rank L.

Note that${{\Xi \quad \left( {M\quad \left( {\Delta \quad c} \right)} \right)} = \begin{bmatrix}{\beta \quad \left( {\Delta \quad {\overset{\_}{c}}_{1}^{\prime}} \right)\quad \beta \quad (M)} \\\vdots \\{\beta \quad \left( {\Delta \quad {\overset{\_}{c}}_{l}^{\prime}} \right)\quad \beta \quad (M)} \\{\beta \quad \left( {\Delta \quad {\overset{\_}{c}}_{l + 1}^{\prime}} \right)\quad \beta \quad (M)} \\\vdots \\{\beta \quad \left( {\Delta \quad {\overset{\_}{c}}_{L}^{\prime}} \right)\quad \beta \quad (M)}\end{bmatrix}},$

where Δ{overscore (c)}_(i)=2Δ{overscore (c)}′_(i) for i>l. Suppose thereare coefficients a₁, a₂, . . . , a_(L) ε such that $\begin{matrix}{0 = \quad {{{a_{1} \cdot \beta}\quad \left( {\Delta \quad {\overset{\_}{c}}_{1}} \right)\quad \beta \quad (M)} \oplus \ldots \oplus {{a_{l} \cdot \beta}\quad \left( {\Delta \quad {\overset{\_}{c}}_{l}} \right)\quad \beta \quad (M)} \oplus {{a_{l + 1} \cdot \beta}\quad \left( {\Delta \quad {\overset{\_}{c}}_{l + 1}^{\prime}} \right)\quad \beta \quad (M)}\quad \oplus}} \\{\quad {\ldots \oplus {{a_{L} \cdot \beta}\quad \left( {\Delta \quad {\overset{\_}{c}}_{L}^{\prime}} \right)\quad \beta \quad (M)}}} \\{= \quad \left\lbrack {{a_{1}\quad \beta \quad \left( {\Delta \quad {\overset{\_}{c}}_{1}} \right)} \oplus \ldots \oplus {a_{l}\quad \beta \quad \left( {\Delta \quad {\overset{\_}{c}}_{l}} \right)} \oplus {a_{l + 1}\quad \beta \quad \left( {\Delta \quad {\overset{\_}{c}}_{l + 1}^{\prime}} \right)} \oplus} \right.} \\{\left. \quad {\ldots \oplus {a_{L}\quad \beta \quad \left( {\Delta \quad {\overset{\_}{c}}_{L}^{\prime}} \right)}} \right\rbrack \quad \beta \quad {(M).}}\end{matrix}$

Then, since β(M) is nonsingular,

a ₁β(Δ{overscore (c)}₁)⊕ . . . ⊕a _(l)β(Δ{overscore (c)}_(l))⊕a_(l+1)β(Δ{overscore (c)}′_(l+1))⊕ . . . ⊕a _(L)β(Δ{overscore(c)}′_(L))=0.

But, by hypothesis, Ξ(Δc) is of full rank. Hence, a₁=a₂= . . . =a_(L)=0,and therefore Ξ(M(Δc)) is also of full rank L as required.

As in the binary case, the transformation theorem implies that certainconcatenated coding schemes preserve the full spatial diversity of aspace-time code. Finally, the results in Section 2.4 regarding thespecial cases of L=2 and 3 for BPSK codes also lift to full diversityspace-time codes for QPSK modulation.

3.2 Dyadic Construction

Two BPSK space-time codes can be directly combined as in a dyadicexpansion to produce a ₄-valued space-time code for QPSK modulation. Ifthe component codes satisfy the BPSK binary rank criterion, thecomposite code will satisfy the QPSK binary rank criterion. Such codesare also of interest because they admit low complexity multistagedecoders based on the underlying binary codes.

Theorem 32 Let and be binary L×n space-time codes satisfying the BPSKbinary rank criterion. Then the ₄-valued space-time code =+2 is an L×nspace-time code that satisfies the QPSK binary rank criterion and thus,for QPSK modulation, achieves full spatial diversity L.

Proof: Let z₁ =a ₁+2 b ₁ and z₂ =a ₂+2 b ₂ be code words in , with a₁,a₂ε and b₁, b₂ ε. Then the ₄ difference between the two code words is

Δz=Δa+2∇a+2Δb,

where Δa=a ₁ ⊕a ₂ , Δb=b ₁ ⊕b ₂, and ∇a=(1⊕a ₁) ⊙ a ₂. In the latterexpression, 1 denotes the all-one matrix and └ denotes componentwisemultiplication. The modulo 2 projection is β(Δz)=Δa, which isnonsingular and equal to Ξ(Δz) unless Δa=0. In the latter case, Δa=0, sothat Δz=2Δb. Then Ξ(Δz)=Δb, which is nonsingular unless Δb=0.

3.3 Mapping Codes to Space-time Codes

Let C be a linear code of length n over ₄. For any code word {overscore(c)}, let w_(i)({overscore (c)}) denote the number of times the symbol iε₄ appears in {overscore (c)}. Furthermore, let w_(i)(C) denote themaximum number of times the symbol i ε₄ appears in any non-zero codeword of C.

The following theorem is a straightforward generalization of the(d_(min), d_(max)) upper bound on achievable spatial diversity forBPSK-modulated codes.

Theorem 33 Let C be a linear code of length n over ₄. Then, for any QPSKtransmission format, the corresponding space-time code achieves spatialdiversity at most

L≦min{n−w ₀(C),n−max{w ₁(C),w ₂(C), w ⁻¹(C)}+1}.

Proof: The same argument applies as in the BPSK case.

It is also worth pointing out that the spatial diversity achievable by aspace-time code is at most the spatial diversity achievable by any ofits subcodes. For a linear ₄-valued code C, the code 2C is a subcodewhose minimum and maximum Hamming weights among non-zero code words aregiven by${{d_{\min}\quad \left( {2C} \right)} = {\min\limits_{\overset{\_}{c} \in C}\left\{ {{w_{1}\quad \left( \overset{\_}{c} \right)} + {w_{- 1}\quad \left( \overset{\_}{c} \right)}} \right\}}},{{d_{\max}\quad \left( {2C} \right)} = {\max\limits_{\overset{\_}{c} \in C}{\left\{ {{w_{1}\quad \left( \overset{\_}{c} \right)} + {w_{- 1}\quad \left( \overset{\_}{c} \right)}} \right\}.}}}$

Thus, the following result is produced.

Proposition 34 Let C be a linear code of length n over ₄. Then, for anyQPSK transmission format, the corresponding space-time code achievesspatial diversity at most

L≦min{d _(min)(2C),n−d _(max)(2C)+1}.

4 ANALYSIS OF EXISTING SPACE-TIME CODES 4.1 TSC Space-time Trellis Codes

Investigation of the baseband rank and product distance criteria for avariety of channel conditions is known, as well as a small number ofhandcrafted codes for low levels of spatial diversity to illustrate theutility of space-time coding ideas. This investigation, however, has notpresented any general space-time code designs or design rules of wideapplicability.

For L=2 transmit antennas, four handcrafted ₄space-time trellis codes,containing 4, 8, 16, and 32 states respectively, are known which achievefull spatial diversity. The 4-state code satisfies simple known designrules regarding diverging and merging trellis branches, and therefore isof full rank, but other codes do not. These other codes require a moreinvolved analysis exploiting geometric uniformity in order to confirmthat full spatial diversity is achieved. The binary rank criterion forQPSK-modulated space-time codes of the present invention, however,allows this determination to be done in a straightforward manner. Infact, the binary analysis shows that all of the handcrafted codes employa simple common device to ensure that full spatial diversity isachieved.

Convolutional encoder block diagrams for ₄ codes of are shown in FIG. 5.The 4-state and 8-state codes are both linear over ₄, with transferfunction matrices G₄(D)=[D 1] and G₈(D)=[D+2D² 1+2D²], respectively. Byinspection, both satisfy the QPSK binary rank criterion of the presentinvention and therefore achieve L=2 spatial diversity.

The 16-state and 32-state codes are nonlinear over ₄. In this case, thebinary rank criterion of the present invention is applied to alldifferences between code words. For the 16-state code, the code wordmatrices are of the following form: $c = {\begin{bmatrix}{x_{1}\quad (D)} \\{x_{2}\quad (D)}\end{bmatrix} = {{\begin{bmatrix}{D + {2D^{2}}} & 0 \\{1 + {2D}} & {2D^{2}}\end{bmatrix}\begin{bmatrix}{z\quad (D)} \\{\alpha \quad \left( {z\quad (D)} \right)}\end{bmatrix}}.}}$

For the 32-state code, the code words are given by $c = {\begin{bmatrix}{x_{1}\quad (D)} \\{x_{2}\quad (D)}\end{bmatrix} = {{\begin{bmatrix}{D + {2D^{2}} + {2D^{3}}} & {3D^{2}} \\{1 + D + {2D^{3}}} & {3D^{2}}\end{bmatrix}\begin{bmatrix}{z\quad (D)} \\{\alpha \quad \left( {z\quad (D)} \right)}\end{bmatrix}}.}}$

Due to the initial delay structure (enclosed by dashed box in FIG. 5)that is common to all four code designs, the first unit inputz_(t)=±1—or first nonzero input z_(t)=2 if z(D) consists only ofmultiples of 2—results in two consecutive columns that are multiples of[0 1]^(T) and [1 x ]^(T), where x ε₄ is arbitrary. The only exceptionoccurs in the case of the 32-state code when the first ±1 is immediatelypreceded by a 2. In this case, the last nonzero entry results in acolumn that is a multiple of [1±1]^(T). Hence, the Ψ-projection of thecode word differences is always of full rank. By the QPSK binary rankcriterion of the present invention, all four codes achieve full L=2spatial diversity.

For L=4 transmit antennas, the full-diversity space-time codecorresponding to the linear ₄-valued convolutional code with transferfunction G(D)=[1 D D² D³ ] is known as a simple form of repetition delaydiversity. As noted in Theorem 13, this design readily generalizes tospatial diversity levels L>4 in accordance with the general designcriteria of the present invention. The stacking and relatedconstructions discussed above, however, provide more generalfull-diversity space-time codes for L≧2.

4.2 GFK Space-time Trellis Codes

For all L≧2, it is known that trellis-coded delay diversity schemesachieve full spatial diversity with the fewest possible number oftrellis states. As a generalization of the TSC simple design rules forL=2 diversity, the concept of zeroes symmetry to guarantee full spatialdiversity for L≧2 is known.

A computer search has been undertaken to identify space-time trelliscodes of full diversity and good coding advantage. A table of best knowncodes for BPSK modulation is available which covers the cases of L=2, 3,and 5 antennas. For QPSK codes, the table covers only L=2. The 4-stateand 8-state QPSK codes provide 1.5 dB and 0.62 dB additional codingadvantage, respectively, compared to the corresponding TSC trelliscodes.

All of the BPSK codes satisfy the zeroes symmetry criterion. Sincezeroes symmetry for BPSK codes is a very special case of satisfying thebinary rank criterion of the present invention, all of the BPSK codesare special cases of the more general stacking construction of thepresent invention.

The known QPSK space-time codes are different. Some of the QPSK codessatisfy the zeroes symmetry criterion; some do not. Except for thetrivial delay diversity code (constraint length ν=2 with zeroessymmetry), all of them are nonlinear codes over ₄ that do not fall underany of our general constructions.

The QPSK code of constraint length ν=2 without zeroes symmetry consistsof the code words c satisfying${{c\quad (D)^{T}} = {\left( {a\quad (D)\quad b\quad (D)} \right)\begin{bmatrix}1 & {2D} \\{2D} & {1 + {2D}}\end{bmatrix}}},$

where a(D) and b(D) are binary information sequences and forsimplicity+is used instead of ⊕₄ to denote modulo 4 addition. The₄-difference between two code words c₁ and c₂, corresponding to inputsequences (a₁(D) b₁(D)) and (a₂(D) b₂(D)), is given by${{\Delta \quad c} = \begin{bmatrix}{{{\Delta \quad a\quad (D)} + {2{\nabla a}\quad (D)} + {2D\quad \Delta \quad b\quad (D)}}\quad} \\{{2D\quad \Delta \quad a\quad (D)} + {\Delta \quad b\quad (D)} + {2{\nabla b}\quad (D)} + {2D\quad \Delta \quad b\quad (D)}}\end{bmatrix}},$

where Δa(D)=a ₁(D)⊕a ₂(D), ∇a(D)=(1⊕a ₁(D))⊙a ₂(D), and so forth. Here ⊙denotes componentwise multiplication (coefficient by coefficient).

Note that the ₄-difference Δc is not a function of the binarydifferences Δa(D) and Δb(D) alone but depends on the individual inputsequences a(D) and b(D) through the terms ∇a(D) and ∇b(D). If Δa(D)=a ₀+a ₁D+a ₂D²+ . . . +a_(N)D^(N), (a_(i) ε), then

Δa(D)+2∇a(D)=±a ₀ ±a ₁ D±a ₂ D ² ± . . . ±a _(N) D ^(N),

for some suitable choice of sign at each coefficient.

Projecting the code word difference Δc modulo 2 gives${{\beta \quad \left( {\Delta \quad c} \right)} = \begin{bmatrix}{\Delta \quad a\quad (D)} \\{\Delta \quad b\quad (D)}\end{bmatrix}},$

which is nonsingular unless either (i) Δa(D)=0, Δb(D)≠0; (ii) Δb(D)=0,Δa(D)≠0; or (iii) Δa(D)=Δb(D)≠0. For case (i), one finds that${{\Xi \quad \left( {\Delta \quad c} \right)} = {\Delta \quad b\quad {(D)\quad\begin{bmatrix}D \\1\end{bmatrix}}}},$

which is nonsingular. For case (ii),${{\Xi \quad \left( {\Delta \quad c} \right)} = {\Delta \quad b\quad {(D)\quad\begin{bmatrix}1 \\D\end{bmatrix}}}},$

which is also nonsingular. Finally, in case (iii), $\begin{matrix}{{\Delta \quad c} = {\begin{bmatrix}{{{\Delta \quad a\quad (D)} + {2{\nabla a}\quad (D)} + {2D\quad \Delta \quad a\quad (D)}}\quad} \\{{\Delta \quad a\quad (D)} + {2{\nabla a}\quad (D)}}\end{bmatrix}.}} & (11)\end{matrix}$

Thus, the t-th column of Δc is given by${\overset{\_}{h}}_{t} = {\begin{bmatrix}{{\Delta \quad a_{t}} + {2{\nabla a_{t}}} + {2\quad \Delta \quad a_{t - 1}}} \\{{\Delta \quad a_{t}} + {2{\nabla a_{t}}}}\end{bmatrix}.}$

Consider the first k for which Δa_(k)=1 and Δa_(k+1)=0 (guaranteed toexist since the trellis is terminated). Then the k-th and (k+1)-thcolumns of Δc are {overscore (h)}_(k)=[±1 ±1]^(T) and {overscore(h)}_(k+1)=[2 0]^(T), respectively. Thus, Ψ(Δc) is nonsingular, and theQPSK binary rank criterion is satisfied. Note that it is the extra delayterm in the upper expression of equation (11) that serves to guaranteefull spatial diversity.

The QPSK code of constraint length ν=3 with zeroes symmetry consists ofthe code words c satisfying${c\quad (D)^{T}} = {{\left( {a\quad (D)\quad b\quad (D)} \right)\quad\begin{bmatrix}{1 + {3D}} & {D + D^{2}} \\2 & {2D}\end{bmatrix}}.}$

For this code, the binary rank analysis is even simpler. The projectionmodulo 2 of the ₄ difference Δc between two code words is given by${{\beta \quad \left( {\Delta \quad c\quad (D)} \right)} = {\Delta \quad a\quad {(D)\begin{bmatrix}{1 \oplus D} \\{D \oplus D^{2}}\end{bmatrix}}}},$

which is nonsingular unless Δa(D)=0 and Δb(D)≠0. In the latter case,${{\Delta \quad c} = {\Delta \quad b\quad {(D)\begin{bmatrix}2 \\{2D}\end{bmatrix}}}},$

whose Ξ- and Ψ-indicants are nonsingular.

The QPSK code of constraint length ν=3 without zeroes symmetry,consisting of the code words${{c(D)}^{T} = {\left( {{a(D)}{b(D)}} \right)\begin{bmatrix}{1 + {2D^{2}}} & {D + {2D^{2}}} \\{1 + {2D}} & 2\end{bmatrix}}},$

does not satisfy the QPSK<binary rank criterion. When Δa(D)=0 butΔb(D)≠0, the code word difference is${{\Delta \quad c} = \begin{bmatrix}{{\Delta \quad {b(D)}} + {2{\nabla{b(D)}}} + {2D\quad \Delta \quad {b(D)}}} \\{2\Delta \quad {b(D)}}\end{bmatrix}},$

for which Ξ(Δc) and Ψ(Δc) are both singular. The latter can be easilydiscerned from the fact that the second row of Δc is two times the firstrow.

4.3 BBH Space-time Trellis Codes

Another computer search is known for L=2 QPSK trellis codes with 4, 8,and 16 states which is similar to the one discussed above. The resultsof the two computer searches agree regarding the optimal productdistances; but, interestingly, the codes found by each have differentgenerators. This indicates that, at least for L=2 spatial diversity,there is a multiplicity of optimal codes.

All of the BBH codes are non-linear over₄. The 4-state and 16-statecodes consist of the following code word matrices: $\begin{matrix}{{4 - {{state}\text{:}\quad {c(D)}^{T}}} = {\left( {{a(D)}{b(D)}} \right)\begin{bmatrix}{2 + D} & {- D} \\2 & {2 + D}\end{bmatrix}}} \\{{16 - {{state}\text{:}\quad {c(D)}^{T}}} = {{\left( {{a(D)}{b(D)}} \right)\begin{bmatrix}{1 + {2D}} & {2 + D + {2D^{2}}} \\{2 + {2D^{2}}} & {2D}\end{bmatrix}}.}}\end{matrix}$

The analysis showing that these two codes satisfy the QPSK binary rankcriterion is straight-forward and similar to that given for the GFKcodes.

The 8-state BBH code consists of the code word matrices${{c(D)}^{T} = {\left( {{a(t)}{b(t)}} \right)\begin{bmatrix}D & 1 \\{2 + {2D} + {2D^{2}}} & {2 + {2D}}\end{bmatrix}}},$

which expression can be rearranged to give${c(D)} = {{{a(D)}\begin{bmatrix}D \\1\end{bmatrix}} + {2{{{b(D)}\begin{bmatrix}{1 + D + D^{2}} \\{1 + D^{2}}\end{bmatrix}}.}}}$

Whereas the GFK 8-state code does not satisfy the QPSK binary rankcriterion, the BBH 8-state code does and is in fact an example of ourdyadic construction =+2. By inspection, the two binary componentspace-time codes and , with transfer functions${{G_{A}(D)} = {{\begin{bmatrix}D \\1\end{bmatrix}\quad {G_{B}(D)}} = \begin{bmatrix}{1 \oplus D \oplus D^{2}} \\{1 \oplus D^{2}}\end{bmatrix}}},$

respectively, both satisfy the BPSK binary rank criterion.

These results show that the class of space-time codes satisfying thebinary rank criteria is indeed rich and includes, for every casesearched thus far, optimal codes with respect to coding advantage.

4.4 Space-time Block Codes From Orthogonal Designs

Known orthogonal designs can give rise to nonlinear space-time codes ofvery short block length provided the PSK modulation format is chosen sothat the constellation is closed under complex conjugation.

Consider the known design in which the modulated code words are of theform $\begin{bmatrix}x_{1} & x_{2}^{*} \\x_{2} & {- x_{1}^{*}}\end{bmatrix},$

where x₁, x₂ are BPSK constellation points. Assuming the on-axis BPSKconstellation, the corresponding space-time block code consists of allbinary matrices of the form $c = {\begin{bmatrix}a & b \\b & {1 \oplus a}\end{bmatrix}.}$

This simple code provides L=2 diversity gain but no coding gain. Thedifference between two modulated code words has determinant${{\det {\begin{matrix}{\left( {- 1} \right)^{a_{1}} - \left( {- 1} \right)^{a_{2}}} & {\left( {- 1} \right)^{b_{1}} - \left( {- 1} \right)^{b_{2}}} \\{\left( {- 1} \right)^{b_{1}} - \left( {- 1} \right)^{b_{2}}} & {- \left\lbrack {\left( {- 1} \right)^{a_{1}} - \left( {- 1} \right)^{a_{2}}} \right\rbrack}\end{matrix}}} = {- \left( {\left\lbrack {\left( {- 1} \right)^{a_{1}} - \left( {- 1} \right)^{a_{2}}} \right\rbrack^{2} + \left\lbrack {\left( {- 1} \right)^{b_{1}} - \left( {- 1} \right)^{b_{2}}} \right\rbrack^{2}} \right)}},$

which is zero if and only if the two code words are identical (a ₁ =a ₂and b ₁ =b ₂). On the other hand, the corresponding binary difference ofthe unmodulated code words is given by${\Delta \quad c} = {\begin{bmatrix}{a_{1} \oplus a_{2}} & {b_{1} \oplus b_{2}} \\{b_{1} \oplus b_{2}} & {a_{1} \oplus a_{2}}\end{bmatrix}.}$

But, if a ₁ ⊕a ₂ =b _(1⊕) b ₂=1, for example, the difference is${{\Delta \quad c} = \begin{bmatrix}1 & 1 \\1 & 1\end{bmatrix}},$

a matrix that is singular over . Hence, achieves full spatial diversitybut does not satisfy the BPSK binary rank criterion.

EXTENSIONS TO NON-QUASI-STATIC FADING CHANNELS

For the fast fading channel, the baseband model differs from equation(1) discussed in the background in that the complex path gains now varyindependently from symbol to symbol: $\begin{matrix}{y_{t}^{j} = {{\sum\limits_{i = 1}^{n}\quad {{\alpha_{ij}(t)}s_{t}^{i}{\sqrt{E}}_{s}}} + {n_{t}^{j}.}}} & (12)\end{matrix}$

Let code word c be transmitted. In this case, the pairwise errorprobability that the decoder will prefer the alternate code word e to ccan be upper bounded by $\begin{matrix}{{P\left( {\left. c\rightarrow e \right.\left\{ {\alpha_{ij}(t)} \right\}} \right)} \leq \quad \left( \frac{1}{\prod\limits_{t = 1}^{n}\quad \left( {1 + {{{{f\left( {\overset{\_}{c}}_{t} \right)} - {f\left( {\overset{\_}{e}}_{t} \right)}}}^{2}{E_{s}/4}N_{0}}} \right)} \right)^{Lr}} \\{{\leq \quad \left( \frac{\mu \quad E_{s}}{4N_{0}} \right)^{{- d}\quad L_{r}}},}\end{matrix}$

where {overscore (c)}_(t) is the t-th column of c, {overscore (e)}_(t)is the t-th column of e, d is the number of columns {overscore (c)}_(t)that are different from {overscore (e)}_(t), and$\mu = {\left( {\prod\limits_{{\overset{\_}{c}}_{t} \neq {\overset{\_}{e}}_{t}}\quad {{{f\left( {\overset{\_}{c}}_{t} \right)} - {f\left( {\overset{\_}{e}}_{t} \right)}}}^{2}} \right)^{1/d}.}$

The diversity advantage is now dL_(r), and the coding advantage is μ.

Thus, the design criteria for space-time codes over fast fading channelsare the following:

(1) Distance Criterion: Maximize the number of column differencesd=|{t:{overscore (c)}_(t)≠{overscore (e)}_(t)}| over all pairs ofdistinct code words c, e ε, and

(2) Product Criterion: Maximize the coding advantage$\mu = {\left( {\prod\limits_{{\overset{\_}{c}}_{t} \neq {\overset{\_}{e}}_{t}}\quad {{{f\left( {\overset{\_}{c}}_{t} \right)} - {f\left( {\overset{\_}{e}}_{t} \right)}}}^{2}} \right)^{1/d}.}$

over all pairs of distinct code words c, e ε.

Since real fading channels are neither quasi-static nor fast fading butsomething in between, designing space-time codes based on a combinationof the quasi-static and fast fading design criteria is useful.Space-time codes designed according to the hybrid criteria are hereafterreferred to as “smart greedy codes,” meaning that the codes seek toexploit both spatial and temporal diversity whenever available.

A handcrafted example of a two-state smart-greedy space-time trelliscode for L=2 antennas and BPSK modulation is known. This code is aspecial case of the multi-stacking construction of the present inventionapplied to the two binary rate ½ convolutional codes having respectivetransfer function matrices ${{G_{1}(D)} = \begin{bmatrix}{1 \oplus D} \\D\end{bmatrix}},\quad {{G_{2}(D)} = {\begin{bmatrix}1 \\{1 \oplus D}\end{bmatrix}.}}$

The known M-TCM example can also be analyzed using the binary rankcriteria. Other smart-greedy examples are based on traditionalconcatenated coding schemes with space-time trellis codes as innercodes.

The general |||, ||⊕|, de-stacking, multi-stacking, and concatenatedcode constructions of the present invention provide a large class ofspace-time codes that are “smart-greedy.” Furthermore, the commonpractice in wireless communications of interleaving within code words torandomize burst errors on such channels is a special case of thetransformation theorem. Specific examples of new, more sophisticatedsmart-greedy codes can be easily obtained, for example, by de-stackingor multi-stacking the space-time trellis codes of Table I. These latterdesigns make possible the design of space-time overlays for existingwireless communication systems whose forward error correction schemesare based on standard convolutional codes. The extra diversity of thespatial overlay would then serve to augment the protection provided bythe traditional temporal coding.

6 EXTENSIONS TO HIGHER ORDER CONSTELLATIONS

Direct extension of the binary rank analysis in accordance with thepresent invention to general L×n space-time codes over the alphabet ₂_(^(r)) for 2 ^(r)-PSK modulation with r≧3 is difficult. Special casessuch as 8-PSK codes with L=2, however, are tractable. Thus, known8PSK-modulated space-time codes are covered by the binary rank criteriaof the present invention.

For general constellations, multi-level coding techniques can producepowerful space-time codes for high bit rate applications while admittinga simpler multi-level decoder. Multi-level PSK constructions arepossible using methods of the present invention. Since at each levelbinary decisions are made, the binary rank criteria can be used inaccordance with the present invention to design space-time codes thatprovide guaranteed levels of diversity at each bit decision.

To summarize, general design criteria for PSK-modulated space-time codeshave been developed in accordance with the present invention, based onthe binary rank of the unmodulated code words, to ensure that fullspatial diversity is achieved. For BPSK modulation, the binary rankcriterion provides a complete characterization of space-time codesachieving full spatial diversity when no knowledge is availableregarding the distribution of ±signs among the baseband differences. ForQPSK modulation, the binary rank criterion is also broadly applicable.The binary design criteria significantly simplify the problem ofdesigning space-time codes to achieve full spatial diversity. Much ofwhat is currently known about PSK-modulated space-time codes is coveredby the design criteria of the present invention. Finally, several newconstruction methods of the present invention are provided that aregeneral. Powerful exemplary codes for both quasi-static and time-varyingfading channels have been identified based on the construction of thecurrent invention and the exemplary set of convolutional codes of TableI.

What is claimed is:
 1. A communication system comprising: a channelencoder for encoding information symbols with an error control code forproducing code word symbols; a spatial formatter for parsing theproduced code word symbols to allocate the symbols to a presentationorder among a plurality of antenna links; and a phase shift keyingmodulator for mapping the parsed code word symbols onto constellationpoints from a discrete complex-valued signaling constellation accordingto binary projections to achieve spatial diversity, wherein said channelencoder comprises a mobile channel encoder producing the code wordsymbols having length of a multiple N of the number of said plurality ofantenna links L.
 2. A communication system as recited in claim 1comprising: a framer for segmenting transmit data blocks into fixedframe lengths for generating information symbols from a discretealphabet of symbols; and data terminal equipment (DTE) coupled to saidframer for communicating digital cellular data blocks.
 3. Acommunication system as recited in claim 2 wherein said digital cellulardata terminal equipment comprises Code Division Multiple Access (CDMA)systems.
 4. A communication system as recited in claim 2 wherein saiddigital cellular data terminal equipment comprises Time DivisionMultiple Access (TDMA) systems.
 5. A communication system as recited inclaim 1, wherein said spatial formatter parses the length N of theproduced code word symbols among L of the transmit antennas.
 6. Acommunication system comprising: a channel encoder for encodinginformation symbols with an error control code for producing code wordsymbols; a spatial formatter for parsing the produced code word symbolsto allocate the symbols to a presentation order among a plurality ofantenna links; and a phase shift keying modulator for mapping the parsedcode word symbols onto constellation points from a discretecomplex-valued signaling constellation according to binary projectionsto achieve spatial diversity, wherein the channel encoder and thespatial formatter utilize a class of space-time codes satisfying abinary rank criteria.
 7. A communication system comprising: a channelencoder for encoding information symbols with an error control code forproducing code word symbols; a spatial formatter for parsing theproduced code word symbols to allocate the symbols to a presentationorder among a plurality of antenna links; and a phase shift keyingmodulator for mapping the parsed code word symbols onto constellationpoints from a discrete complex-valued signaling constellation accordingto binary projections to achieve spatial diversity, wherein thecombination of channel encoder and spatial formatter are provided with astacking space-time code construction.
 8. A communication system asrecited in claim 1, wherein binary phase shift keying (BPSK) modulationis used and a space-time code is based on formatting the output ofconvolutional channel encoder for presentation across a plurality oftransmit antennas.
 9. A communication system comprising: a channelencoder for encoding information symbols with an error control code forproducing code word symbols; a spatial formatter for parsing theproduced code word symbols to allocate the symbols to a presentationorder among a plurality of antenna links; and a phase shift keyingmodulator for mapping the parsed code word symbols onto constellationpoints from a discrete complex-valued signaling constellation accordingto binary projections to achieve spatial diversity, wherein aconcatenated space-time code is used in which the outer code is used tosatisfy a binary rank criteria and multiple inner codes are used toencode the transmitted information from a plurality of transmitantennas.
 10. A communication system comprising: a channel encoder forencoding information symbols with an error control code for producingcode word symbols; a spatial formatter for parsing the produced codeword symbols to allocate the symbols to a presentation order among aplurality of antenna links; and a phase shift keying modulator formapping the parsed code word symbols onto constellation points from adiscrete complex-valued signaling constellation according to binaryprojections to achieve spatial diversity, wherein a concatenatedspace-time code is used in which the inner code is composed of a channelencoder and a spatial formatter designed to satisfy a binary rankcriteria.
 11. A communication system comprising: a channel encoder forencoding information symbols with an error control code for producingcode word symbols; a spatial formatter for parsing the produced codeword symbols to allocate the symbols to a presentation order among aplurality of antenna links; and a phase shift keying modulator formapping the parsed code word symbols onto constellation points from adiscrete complex-valued signaling constellation according to binaryprojections to achieve spatial diversity, wherein the combination ofchannel encoder and spatial formatter are covered by a multi-stackingconstruction.
 12. A communication system comprising: a channel encoderfor encoding information symbols with an error control code forproducing code word symbols; a spatial formatter for parsing theproduced code word symbols to allocate the symbols to a presentationorder among a plurality of antenna links; and a phase shift keyingmodulator for mapping the parsed code word symbols onto constellationpoints from a discrete complex-valued signaling constellation accordingto binary projections to achieve spatial diversity, wherein quadraturephase shift keying (QPSK) modulation is used and a space-time code iscovered by a stacking, multi-stacking, or de-stacking constructions forQPSK.
 13. A communication system as recited in claim 1, wherein QPSKmodulation is used and a space-time code is based on formatting anoutput of a linear convolutional code over the ring of integers modulo 4for presentation across a plurality of transmit antennas.
 14. Acommunication system comprising: a channel encoder for encodinginformation symbols with an error control code for producing code wordsymbols; a spatial formatter for parsing the produced code word symbolsto allocate the symbols to a presentation order among a plurality ofantenna links; and a phase shift keying modulator for mapping the parsedcode word symbols onto constellation points from a discretecomplex-valued signaling constellation according to binary projectionsto achieve spatial diversity, wherein QPSK modulation is used and aspace-time code employs a dyadic construction.
 15. A communicationsystem comprising: a channel encoder for encoding information symbolswith an error control code for producing code word symbols, a spatialformatter for parsing the produced code word symbols to allocate thesymbols to a presentation order among a plurality of antenna links; anda phase shift keying modulator for mapping the parsed code word symbolsonto constellation points from a discrete complex-valued signalingconstellation according to binary projections to achieve spatialdiversity, wherein multi-level coded modulation and multi-stage decodingare used and a binary space-time code employed in each level belongs tothe class of codes that satisfy the binary rank criterion or any of theconstructions.
 16. A communication system comprising: a channel encoderfor encoding information symbols with an error control code forproducing code word symbols; a spatial formatter for parsing theproduced code word symbols to allocate the symbols to a presentationorder among a plurality of antenna links; and a phase shift keyingmodulator for mapping the parsed code word symbols onto constellationpoints from a discrete complex-valued signaling constellation accordingto binary projections to achieve spatial diversity, wherein M-ary PSKmodulation, M=8 or more, is used and a space-time code belongs to aclass satisfying the binary rank criteria.
 17. A communication systemcomprising: a channel encoder for encoding information symbols with anerror control code for producing code word symbols; a spatial formatterfor parsing the produced code word symbols to allocate the symbols to apresentation order among a plurality of antenna links; and a phase shiftkeying modulator for mapping the parsed code word symbols ontoconstellation points from a discrete complex-valued signalingconstellation according to binary projections to achieve spatialdiversity, wherein graphical space-time codes are designed such that thecode generating matrix satisfies the stacking, multi-stacking, orde-stacking construction.
 18. A communications method comprising:encoding information symbols with an underlying error control code toproduce code word symbols; parsing the produced code word symbols toallocate the symbols in a presentation order to a plurality of antennalinks, wherein the code word symbols have length of a multiple N of thenumber of said plurality of antenna links L; and mapping the parsed codeword symbols onto constellation points from a discrete complex-valuedsignaling constellation, wherein the symbols are modulated fortransmission over a plurality of transmit antennas corresponding to theantenna links.
 19. A method as recited in claim 18 wherein said encodingand parsing steps are performed with a space-time encoder having achannel encoder and a space-time formatter.
 20. A communications systemcomprising: means for encoding the generated information symbols with anunderlying error control code to produce code word symbols; means forparsing the produced code word symbols to allocate the symbols in apresentation order to a plurality of antenna links, wherein the codeword symbols have length of a multiple N of the number of said pluralityof antenna links L; and means for mapping the parsed code word symbolsonto constellation points from a discrete complex-valued signalingconstellation, wherein the symbols are modulated for transmission over aplurality of transmit antennas corresponding to the antenna links.
 21. Acommunication system as recited in claim a 20 wherein said parsing meanscomprises a spatial formatter for parsing the length N of the producedcode word symbols among the L antennas.
 22. A communication system asrecited in claim 20 wherein said encoding and parsing means comprise aspace-time encoder having a channel encoder and a space-time formatter.